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Development of 
Gear Technology and 
Theory of Gearing 



\zz< 





by Faydor L. Litvin 



RESEARCH LABORATORY 



National Aeronautics and 
Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135 




NASA Reference Publication 1406 ARL-TR-1500 

U.S. ARMY 







RESEARCH LABORATORY 



Development of 
Gear Technology and 
Theory of Gearing 



Faydor L. Litvin 

University of Illinois at Chicago, Chicago, Illinois 



National Aeronautics and 
Space Administration 

Lewis Research Center 



Acknowledgments 

The author expresses his deep gratitude to the individuals and institutions for the historical information about the 
development of gear theory and technology and to those companies that have supported the research projects accom- 
plished at the Gear Research Laboratory of the University of Illinois. 

From the United States: 

Allison Engine Company: Gene Pfaffenberger, Supervisor, and Matt Hawkins, Development Engineer; Army 
Research Laboratory (NASA Lewis Research Center): Dr. Robert Handschuh, Dr. David G. Lewicki, and Dr. Robert Bill; 
Paul Baxter, the son of Meriwether L. Baxter; Buckingham Associates Incorporated: Eliot K. Buckingham, President; 
Cone Drive Textron: Jerry B. Hagaman, Director of Engineering, and Duane Gilbert, Manager of Sales and Marketing; 
Dana Corporation, Spicer Axle Division: John Hickey, Director of Engineering, and Dr. Wei-Jiung Tsung, Gear Research 
Coordinator; Darie W. Dudley; Emerson Power Transmission Corporation: Larry Spiers, Vice President of Engineering, 
and Charies A. Libourel, Manager of Research and Development; Fellows Gear Shaper Company: Lawrence E. Greenip, 
Jr.,Vice President; Ford Motor Company: Ronald A. Andrade, Chief Engineer, and Thomas M. Sep, Senior Technical 
Specialist; The Gleason Foundation: John B. Kodweis, Vice President of Administration; The Gleason Works: Gary J. 
Kimmet, Vice President of Regional Operations; Dr. Hermann J. Stadtfeld, Vice President of Research and Development; 
and Theodore J. Krenzer, Director of Research and Development (retired); Margot Jerrad. the daughter of Dr. Hillei 
Poritsky; Mary Bell Kluge, the granddaughter of Samuel I. Cone; McDonnell Douglas Helicopter Systems: Robert J. 
King, Project Manager; Terrell W. Hansen, Department Manager; and Roger J. Hunthausen, Technology Development 
Manager; NASA Lewis Research Center: Dr. John J. Coy, Chief, Mechanical Components Branch, and Dennis P. Townsend; 
Reishauer Corporation: George I. Wyss, President, U.S. Division; Veikko Saari and Virginia Saari Jensen, the brother and 
daughter of Oliver E. Saari; Richard L. Thoen, gear consultant 

From Germany: 

Klingelnberg Sohne: Deither Klingelnberg, general partner; Maschinenfabrik Lorenz GMBH: Klaus Felten, Doctor of 
Engineering and Managing Director; Munich Technical University: Prof. Hans Winter, Doctor of Engineering; Hermann 
Pfauter GMBH & Co.: Gunter R. Schmidt 

From Russia: 

Kurgan Technical University: Dr. and Prof. Maks L. Erikhov; St. Peterburg Polytechnic University: Prof. Georgie A. 
Smimov 

From England: 

David Brown Group PLC: John S. Hudson 

From Switzerland: 

Maag Holding AG: Samuel Gartmann, Chairman of the Board; Oeriikon Geartec AG: Urs Koller, President 



Available from 

NASA Center for Aerospace Information National Technical Information Service 

800 Elkridge Landing Road 5287 Port Royal Road 

Linthicum Heights, MD 21090-2934 Springfield, VA 22100 

Price Code: A06 Price Code: A06 



NASA RP-1406 



Foreword 



Aeronautics, a principal research area for NASA from its inception as NACA, has also been the primary focus 
for the NASA Lewis Research Center since the 1940's, In the 1970's, the Army established drive systems 
technology as part of its research to support its ever increasing use of helicopters. This effort has continued to 
the present and is now a shared activity of the Army and NASA because of the need for civilian applications. 

NASA and Army goals for this research are similar and emphasize reduced weight, noise, and cost with 
increased safety, life, and reliability. Meeting these competing goals requires that improvements be made in 
the way components are designed and manufactured. These goals are being met by in-house programs, 
contracts, and university grants, through which the talents of world-class university researchers have an impact 
on aerospace technology and products of the future. 

Dr. Faydor L. Litvin, one of the innovators in the gear geometry field for many decades, best exemplifies the 
effective use of NASA funds for research. Dr. Litvin' s methods and theories have been the catalyst to change 
the design and manufacture of gears so as to achieve major operational improvements in helicopter gear 
systems. He effected these improvements by applying principles based on the geometry of meshing gear 
surfaces to correct many problems associated with alignment and manufacturing errors that shift bearing contact 
and cause transmission errors. 

This book presents recent developments in the theory of gearing and the modifications in gear geometry 
necessary to improve the conditions of meshing. Highlighted are low-noise gear drives that have a stable contact 
during meshing and a predesigned parabolic transmission error function that can handle misalignment during 
operation without sacrificing the low-noise aspects of operation. 

This book also provides a comprehensive history of the development of the theory of gearing through 
biographies of major contributors to this field. The author's unique historical perspective was achieved by 
assiduous research into the lives of courageous, talented, and creative men who made significant contributions 
to the field of gearing. Very often they came from humble backgrounds, sought an education in the face of great 
obstacles, made personal sacrifices to attain goals, and worked hard for many years to fulfill their creative 
aspirations. The task of accumulating information about these men was extremely difficult, for many were 
deceased and facts existed only in family records, library archives, and their companies' files. Perhaps the most 
significant indication of this difficult task is the collection of portraits, many of which were obtained from family 
albums held by descendents many generations later. This collection is unique and exists nowhere else. 

I believe that this book will be useful to students, engineers, and researchers who work in gearing and that 
it will help them to appreciate the genius of those who were pioneers in this field. 



John J. Coy 

NASA Lewis Research Center 



NASA RP-1406 iii 



PAGE INTENTIONALLY LEFT BLANK 



Preface 



There yet remains but one concluding tale 
And then this chronicle of mine is ended. . .—A. S. Pushkin, "Boris Godunov," p. 19 

This book consists of three chapters: chapter 1 presents the developments in the theory of gearing; chapter 2, 
the geometry and technology of gears; and chapter 3, the biographies of the inventors, scientists, and founders 
of gear companies. The goal of the third chapter is to credit the contributions made by our predecessors and to 
combine the separate historical pieces of the development of gear technology and gearing theory. 

Although the author has tried to be objective in judging the history of the development of gear technique, it 
is not a certainty that he has achieved his desire. If this happened, it was not deliberate. 

The history of gear development was the subject of research by H.-Chr. Graf v. Seherr-Thoss (1965), Darle 
W. Dudley (1969), and Dr. Hermann J. Stadtfeld (1993). In the present book, the author tried to complement 
these previous publications with new materials and hopes that the reader will find it enlightening. 

The history of developments in any area, including gear technology and theory, is the history of creativity, 
which has often gone unrecognized during one's lifetime. The aspiration to create is a passion that enriches the 
life but requires unconditional devotion. Usually, creativity is associated with the arts (music, literature, 
painting), possibly because they have the greatest influence on our emotions. However, we do not realize the 
extent to which this passion conquers the daily activities of many in all levels of society. The desire of gifted 
persons to create is the driving force in their lives, bringing them joy and suffering and often no fame. For, Fame, 
a capricious goddess, does not award in the proper time and may not award at all. 

My sympathy is for those who failed to achieve recognition for their accomplishments, and I share 
Dostoyevsky's philosophy that suffering is necessary for spiritual achievement, but the price to be paid is 
sometimes too high. However, an individual who gives his heart to create should not look for fame. This was 
expressed with great emotion by Pasternak (1960) in his famous verse "To Be That Famous Is Hardly 
Handsome": 

Creation 's aim— yourself to give, 

Not loud success, appreciation. 

To mean round nothing— shames to live. 

On all men 's lips an empty sermon. 

I sympathize with the heroes of Pasternak's verse. 

Although J. Henry Fabre' s area of activity was different from the author's, he is an example of someone whose 
life deeply touched the author. Fabre (1949) was a famous entomologist who exemplified the unconditional 
devotion that creativity required. He was very poor, and struggled to support his family. However, every day 
he was in the field to observe the social life and habits of insects — the subject of his research. He made a modest 
confession that all he wanted was "a bit of land, oh, not so very large, but fenced in" where he could study insects. 
Fabre puzzled not only his neighbors but also his contemporaries by spending his time studying insects, but 
through his devotion to this subject, he became the founder of entomology . Unfortunately, Fabre became known 
only at the end of his long life (1823 to 1915), and he received a pension for only his last 5 years. 



NASA RP-1 406 



Another example is the Italian mechanic and clockmaker Juanelo Torriano (1501 to 1575), the inventor of 
the first known gear cutting machine (Dudley, 1969). He spent 20 years on this project and paid a high price 
for his intensive work. He was sick twice and almost died before he successfully created his manually operated 
machine. Maybe his reward was self-satisfaction. 

The theory and technology of gearing is a narrow branch of science, but the author believes that what was 
said about the creative process holds true here. Who knows how many sleepless nights an inventor, a scientist, 
or the founder of a new gear company had? How often did they ponder Hamlet' s question "To be or not to be?" 
Recognizing the achievements of one's predecessors is the best way to honor their work. This is one of the goals 
of this book written in memory of the founders of the gear industry and gear technology and theory. 

The author is very thankful for Dr. S. A. Lagutin's and Dr. P.-H. Feng's invaluable help in the preparation 
of this book for publication. The author feels a deep gratitude to the skillful and intellectual members 
of the NASA Lewis publishing department, J. Berkopec, D. Easter, M. Grimes, J. Jindra, A. Crawford, 
N. Mieczkowski, L. Feher, and F. Turner, for their high-quality editorship of the book. 



Faydor L. Litvin 

The University of Illinois 

at Chicago 



NASARP-1406 vi 



Contents 



Acknowledgments ii 

Foreword iii 

Preface v 

Chapter 1 Development of Theory of Gearing 

1.1 Introduction 1 

1 .2 Equation of Meshing 2 

1 .3 Basic Kinematic Relations 4 

1.4 Detection and Avoidance of Singularities of Generated Surface 5 

1 .5 Direct Relations Between Curvatures of Meshing Surfaces: 

Instantaneous Contact Ellipse , 8 

1 .6 Sufficient Conditions for Existence of Envelope to Family of Surfaces 9 

1 .7 Envelope E2 to Contact Lines on Generated Surface Z2 and Edge of Regression 11 

1 .8 Necessary and Sufficient Conditions for Existence of Envelope E| to 

Contact Lines on Generating Surface I^ 16 

1.9 Axes of Meshing 19 

1.10 Two-Parameter Enveloping 28 

1.11 Localization of Contact and Simulation of Meshing of Gear Tooth Surfaces 28 

1.12 Equation of Meshing for Surfaces in Point Contact 30 

1.13 Transition From Surface Line Contact to Point Contact 30 

1.14 Design and Generation of Gear Drives With Compensated Transmission Errors 32 

Appendix A Parallel Transfer of Sliding Vectors 35 

Appendix B Screw Axis of Motion: Axodes 38 

B.l Screw Motion 38 

B.2 Axodes 41 

Appendix C Application of Pluecker's Coordinates and Linear Complex in Theory of Gearing 

C.l Introduction 42 

C.2 Pluecker's Presentation of Directed Line 42 

C.3 Pluecker's Linear Complex for Straight Lines on a Plane 43 

C.4 Application to Screw Motion 44 

C.5 Interpretation of Equation of Meshing of One-Parameter Enveloping Process 47 

C.6 Interpretation of Equations of Meshing of Two-Parameter Enveloping Process 49 

Chapter 2 Development of Geometry and Technology 

2.1 Introduction 53 

2.2 Modification of Geometry of Involute Spur Gears 53 

2.3 Modification of Involute Helical Gears 56 

2.4 Development of Face-Gear Drives 59 

2.5 Development of Geometry of Face-Milled Spiral Bevel Gears 63 



NASARP-1406 



Vll 



2.6 Modification of Geometry of Worm-Gear Drives With Cylindrical Worms 66 

2.7 Face Worm-Gear Drives 71 

2.8 Development of Cycloidal Gearing 73 

Chapter 3 Biographies and History of Developments 

3.1 Introduction 79 

3.2 Johann Georg Bodmer — Inventor, Designer, and Machine Builder (1786-1864) 80 

3.3 Friedrich Wilhelm Lorenz — Doctor of Engineering, h.c. and Medicine, h.c, 

Inventor, and Founder of the Lorenz Company (1842-1924) 81 

3.4 Robert Hermann Pfauter — Inventor and Founder of the Pfauter Company 

(1854-1914) 82 

3.5 Edwin R. Fellows — Inventor and Founder of the Fellows Gear Shaper Company 

(1865-1945) 83 

3.6 Max Maag — Doctor of Engineering, h.c, Inventor, and Founder of the Maag Company 

(1883-1960) 84 

3.7 Earle Buckingham — Professor of Mechanical Engineering, Gear Researcher, and 

Consultant (1887-1978) 85 

3.8 Ernst Wildhaber — Doctor of Engineering, h.c. Inventor, and Consultant for The 

Gleason Works (1893-1979) 86 

3.9 Hillel Poritsky — Doctor of Mathematics, Professor of Applied Mechanics, 

and Consultant (1898-1990) 87 

3.10 Gustav Niemann — Professor and Doctor of Mechanical Engineering (1899-1982) 88 

3.1 1 Meriwether L. Baxter, Jr.— Gear Theoretician (1914-1994) 89 

3.12 Hans Liebherr — Doctor of Engineering, h.c, Designer, and Founder of the 

Liebherr Group Companies (1915-1993) 90 

3.13 Alexander M. Mohrenstein-Ertel — Doctor of Engineering (1913-) 91 

3.14 Darle W. Dudley— Gear Consultant, Researcher, and Author (1917-) 92 

3.15 Oliver E. Saari— Inventor at the Illinois Tool Works (ITW) Spiroid Division (1918-) . . 93 

3.16 History of the Invention of Double-Enveloping Worm-Gear Drives 94 

3.17 History of the David Brown Company 95 

3.18 History of The Gleason Works 96 

3.19 History of the Klingelnberg and Oeriikon Companies 98 

3.20 History of the Reishauer Corporation 99 

3.21 The Development of the Theory of Gearing in Russia 100 

Chaim I. Gochman— Doctor of Applied Mathematics (1851-1916) 101 

Chrisanf F. Ketov — Doctor of Technical Sciences and Professor of Mechanisms 

and Machines (1887-1948) 102 

Nikolai I. Kolchin — Doctor of Technical Sciences and Professor of the Theory of 

Mechanisms and Machines (1894-1975) 103 

Mikhail L. Novikov — Doctor of Technical Sciences and Professor and 

Department Head (1915-1957) 104 

Vladimir N. Kudriavtsev — Doctor of Technical Science and Professor and Head 

of the Department of Machine Elements (1910-1996) 105 

Lev V. Korostelev— Doctor of Technical Sciences and Professor (1923-1978) 106 

References 107 

Index Ill 



NASA RP-1 406 viii 



Chapter 1 

Development of Theory of Gearing 

1.1 Introduction 

The theory of gearing is the branch of science related to differential geometry, manufacturing, design, 
metrology, and computerized methods of investigation. The first developers of the theory of gearing (Olivier 
and Gochman) related it to projective and analytical geometry. Later, with the development of gear technology 
and the application of computers to gearing, researchers modified it to the modem theory of gearing and 
extended its methodology and industrial applications. 

One of the most important problems the theory of gearing considers is the conjugation of profiles in planar 
gearing and surfaces in spatial gearing. De la Hire, Poncelet and Camus (Seherr-Thoss, 1 965) deserve credit for 
developing cycloidal gearing. Euler (1781), in addition to his tremendous contribution to mathematics and 
mechanics, proposed involute gearing that was later found to have a broad application in industry (fig. Ll.l). 
The Swiss government issued currency with 
Euler' s picture and an involute curve to 
celebrate his achievements. Olivier (1842) 
and Gochman ( 1 886) developed the basic 
ideas underlying the conjugation of gear 
tooth surfaces and their generation. N.I. 
Kolchin (1949) applied Gochman' s ideas 
to develop the geometry of modem gear 
drives. 

Willis, Buckingham, Wildhaber, and 
Dudley are very well-known names in 
English-speaking countries. WiUis (1841) 
proposed the law of meshing of planar 
curves. Buckingham* s book ( 1 963) became 
a well-known reference on gears. 
Wildhaber, an inventor who held many 
patents, developed the theory of hypoid 
gear drives and the Revacycle method 
(Wildhaber, 1926, 1946a, 1946b, 1946c, 
1956). Dudley (1943, 1954, 1961, 1962, 
1969, 1984, 1994) was the chief editor of 
the first edition of the Gear Handbook 
the foremost work on gears at that time. 
He became a prominent gear expert and 
well known for his contributions to gear 
technology. 



Figure 1 .1 .1 . — Leonhard Euler. 




NASA RP-1406 



The former U.S.S.R. conducted intensive gear research. In the 1940*s and 1950's, Profs. Chrisanf F. Ketov, 
N. I. Kolchin, V.A. Gavrilenko, and others supervised the research activities of many graduate students. The 
author, one of these students, remembers with gratitude and respect his advisor. Prof. Ketov. A few of the many 
distinguished Russian researchers from this generation are V.N. Kudriavtsev, G.I. Sheveleva, 
L.V. Korostelev, Ya. S. Davidov, and M.L. Erikhov. Several research centers in Western countries have 
contributed to the areas of gear geometry, technology, and dynamics: 

(1) The Gleason Works in Rochester, New York; the NASA Lewis Research Center in Cleveland, Ohio; The 
Ohio State University in Columbus, Ohio; the University of Illinois in Chicago, Illinois (United States) 

(2) The Universities of Munich, Aachen, Stuttgart, and Dresden (Germany) 

(3) Institute de L*Engrenage et des Transmissions and Cetim (France) 

(4) The Universities of Laval and Alberta (Canada) 

Important contributions to the theory of gearing have been provided by 

(1) E.Buckingham (1963), E.Wildhaber (1926,1946, 1956), D.Dudley (1943, 1954, 1961, 1962, 1969, 1984, 
1991), M.Baxter (1961, 1973), T.Krenzer (1981), A. Seireg (1969), G.Michalek (1966), and Y.Gutman 
from the United States 

(2) G.Niemann (1953), G.R.Brandner (1983, 1988),H.Winter,B.Hohn,M. Week, and G.Bar(1991, 1997) 
from Germany 

(3) H. Stadtfeld (1993, 1995) formerly of Switzerland but now in the United States 

(4) G. Henriotte and M. Octrue from France 

(5) C. GosseHn (1995) and J.R. Colboume (1974, 1985) from Canada 

This chapter presents the latest developments in the theory of gearing. These resulted from the work of the 
author and his fellow researchers at the University of Illinois Gear Research Laboratory in Chicago. 

We are on the eve of the 21st century and can expect that technology such as the CNC (computer numerically 
controlled) and CCM (computer coordinate measurement) machines will substantially change existing gear 
geometry and gear technology. 

1.2 Equation of Meshing 

The generation of a gear tooth surface by the tool surface (the surface of a head-cutter, hob, shaper, rack-cutter, 
etc.) and the conjugation of gear tooth surfaces in line contact are based on the concept of the envelope to a family 
of surfaces (curves in two-dimensional space in the case of planar gearing). This topic is related to differential 
geometry and to the theory of gearing. Zalgaller*s book (1975) significantly contributes to the theory of 
envelopes and covers the necessary and sufficient conditions for the envelope's existence. Simplified 
approaches to the solution of these problems have also been developed in the theory of gearing (Litvin, 1968, 
1989). 

In further discussions, we use the notations Zj and Xj for the generating and generated surfaces, respectively. 
The applied coordinate systems are designated by 5p ^j, and Sp which are rigidly connected to Xj, i^, and to 
the fixed frame of the machine (housing) where the axes of rotation of Zj and i^ ^^ located. 

We consider that Xj is represented as 

ri(w,0)GC\ — ix—^^O, {u,6)sE (1.2.1) 

du du 

where (m,0) are surface parameters, and C^ indicates that vector function rj(w,0) has continuous partial 
derivatives of at least the first order. The inequality in (1,2.1) indicates that E^ is a regular surface. 
Using the coordinate transformation from 5j to 52, we obtain the family of surfaces Xj represented in ^2 as 

r2(M,0,0) = [x2(i/,^,0), y2(uA<t>l Z2iu,e,(l>)f (1.2.2) 



NASARP-1406 



where 0is the generalized parameter of motion. When ^is fixed, equation (1.2.2) represents the surface Xjin 
the 52 coordinate system. 

Envelope Z2 is in tangency with all surfaces of the family of surfaces (eq. (1.2.2)), Surface I^ can be 
determined if vector function r2(uA<t>) of the family of surfaces will be complemented by 

/(w,0,0) = O (1.2.3) 

In 1952 in the theory of gearing, equation ( 1 .2.3) had received the term "equation of meshing" (Litvin, 1 952). 
Several alternative approaches have since been proposed for the derivation of equation (1.2.3). 

Approach 1. The method of deriving the equation of meshing was proposed in differential geometry and is 
based on the following considerations: 

(1) Assume that equation (1 .2.3) is satisfied with a set of parameters P(w°, 0^, 0°). It is given that f e C^ and 
that one of the three derivatives (/y,/^,/^, say/^^, is not equal to zero. Then, in accordance with the theorem 
of implicit function system existence (Kom and Kom, 1968), equation (1.2.3) can be solved in the neighbor- 
hood of P = (M^0^,<^°) by function w(0,<^). 

(2) Consider now that surface L^ may be represented by vector function r2(0,0,M(0,0)) and that the tangents 
to 2^ may be represented as 

(3) The normal N^'^ to surface Xj that is represented in 52 is determined as 

N<.. = fxf 0.2., 

The subscript 2 in the designation N^^ indicates that the normal is represented in 52; the superscript (1) 
indicates that the normal to Z^ is considered. 

(4) If the envelope Xj exists, it is in tangency with Zj, and Xj and Xj "^ust have a common tangent plane. A 
tangent plane n^^) ^q 2^ jg determined by the couple of vectors T2 and T|. A tangent plane 11^^ to X, is 
determined by the couple of vectors dr2/du and dr2/d9. Vector T2 lies in plane n^^^ already. Surfaces Xj and 
Xj will have a common tangent plane if vector dr2ld(pd\so lies in U^\ The requirement that vectors 

[dr2^du,dT2ldO and dr2/d<l>) belong to the same plane f FIj J is represented by 

Remember that (1.2.6) is the equation of meshing and that equations (1.2.2) and (1.2.6) considered 
simultaneously represent surface Xj, the envelope to the family of surfaces Xj. 

Approach 2. This approach is based on the following considerations: 

(1) The cross product in equation (1.2.6) represents in 52 the normal to X| (see eq. (1.2.5)). 

(2) The derivative dr2/d<p'\s collinear to the vector of the relative velocity v^^^\ which is the velocity of a 
point of Xj with respect to the coinciding point of X^. This means that equation (1.2.6) yields 

Ny^-vy2)=/(i,,0^0) = O (1.2.7) 

(3) The scalar product in (1 .2.7) is invariant to the coordinate systems 5,, Sj, and 52- Thus 

^\^^'y\^^^=f{uA(p) = (i = l,/,2) (1.2,8) 



NASARP-1406 



The derivation of the equation of meshing becomes more simple if / = 1, or / =/. 

Equation (1.2.8) was proposed almost simultaneously by Dudley and Poritsky, Davidov, Litvin, Shishkov, 
and Saari. Litvin has proven that equation ( 1 .2.8) is the necessary condition for the envelope* s existence (Litvin, 
1952 and 1989). 

The determination of the relative velocity v^^^^ can be accomplished using well-known operations applied 
in kinematics (see appendix A). In the case of the transformation of rotation between crossed axes, an alternative 
approach for determining v^^^^ may be based on the application of the concept of the axis of screw motion 
(appendix B). 

In the case of planar gearing, the derivation of the equation of meshing may be represented as 

T.(l)xvP^^=0 (1.2.9) 



or 



(T/^>xv|^2>)-ky=0, (/ = 1,2,/) (1.2.10) 

where, T^^^ is the tangent to the generating curve, v/^^^ is the sliding velocity, k^. is the unit vector of the z-axis 
(assuming that the planar curves are represented in plane (jc-, y-)). 

In the cases of planar gearing and gearing with intersected axes, the normal to the generating curve (surface) 
at the current point of tangency of the curves (surfaces) passes through ( 1 ) the instantaneous center of rotation 
for planar gearing (first proposed by Willis (1941)), and (2) the instantaneous axis of rotation for gears with 
intersected axes. The derivation of the equation of meshing for gearing with intersected axes is based on 

]kzlL = IlzIL = liZlL (i = l2 f) a 2 1 n 

N^;> N<;) N^;) ' ^ "^^ " ^ 

•*i /(' ^i 

where (X-, K-, Z-) are the coordinates of a current point of the instantaneous axis of rotation; (x-, y-, z-) are the 
coordinates of a current point of the generating (driving) surface; N^\N^/,N^/ are the projections of the 
normal to surface Z|. 



1.3 Basic Kinematic Relations 

Basic kinematic relations proposed in Litvin (1968 and 1989) relate the velocities (infinitesimal displace- 
ments) of the contact point and the contact normal for a pair of gears in mesh. 

The velocity of a contact point is represented as the sum of two components: in the motions with and over 
the contacting surface, respectively. Using the condition of continuous tangency of the surfaces in mesh, we 
obtain 

^(2)^^(1)^^(12) (J31) 

where v/'^ (/ = 1,2) is the velocity of a contact point in the motion over surface E^. Similarly, we can represent 
the relation between the velocities of the tip of the contact normal 

h?>=n(;>+(a)('2)xn) (1.3.2) 

where, h^'^ (/ = 1 ,2) is the velocity of the tip of the contact normal in the motion over the surface (in addition 
to the translational velocity of the unit normal n that does not affect the orientation of n), and ©^ * ^^ is the relative 
angular velocity of gear I with respect to gear 2. 

The advantage in using equations (1.3.1) and (1.3.2) is that they enable the determination of v^^^^ and h^^^ 
without having to use the complex equations of the generated surface L^. 

By applying equations (1.3.1) and (1.3.2) for the solutions of the following most important problems in the 
theory of gearing, the application of the complex equations of X2 has been avoided: 



NASA RP-1406 



Problem 1 : Avoidance of singularities of the generated gear tooth surface 1^ 

Problem 2: Determination of the principal curvatures, the normal curvatures, and the surface torsions of 2^ 

Problem 3: Determination of the dimensions and the orientation of the instantaneous contact ellipse 

1.4 Detection and Avoidance of Singularities of Generated Surface 

Generating tool surface Xj is already free from singularities because the inequality (dr^/du) x (dr^/dd) ^ 
was observed. Tool surface E^ may generate surface I^, which will contain not only regular surface points but 
also singular ones. The appearance of singular points on Z^ is a warning of the possible undercutting of I^ in 
the generation process. 

The discovery of singular points on Xj may be based on the theorem proposed in Litvin (1968): a singular 
point M on surface Zj occurs if at M the following equation is observed: 



v(;>=v<;>+v<'2)=o 



Equation (1.4.1) and the differentiated equation of meshing 

d 



dr 



[me,w)]=o 



yield 



^r, du ^ A-i de ^ ^(12) 
du dt de dt ' 



^du ^d0 
du dt de dt 



^d0 
'd<p dt 



(1.4.1) 
(1.4.2) 

(1.4.3) 
(1.4.4) 



Taking into the account that Z, is a regular surface, we may transform (1.4.3) as 

v2 



{dej [dujdt [de) dt [dej v ■ / 



(1.4.5) 



(1.4.6) 



Equations (1.4.4) to (1.4.6) form an overdetermined system of three linear equations in two unknowns: 
du/dt, d9fdt. The rank of the system matrix is r = 2, which yields 



gl(u,e,<l>) = 



fu 



fe 



u 



Kdu] {du)\dd] \du)^' 

2 



Kde) 
\dd)\du) \de) Kde)^ 



,(12)' 



,(12) 



= 



(1.4.7) 



We have assigned in v^'^^ that d^dt = 1 rad/sec. 

Equations/= and ^| = allow us to determine on X, curve L,, which is formed by surface "regular" points 
that generate "singular" points on L^. Using the coordinate transformation from Sy to Sj, we may determine 
curve L2. which is formed by singular points on surface I^. To avoid the undercutting of 2^, it is sufficient to 
limit I2 and to eliminate L2 while designing the gear drive. 

It is easy to verify that the theorem proposed above yields a surface normal N^^^ equal to zero at the surface 
I2 singular point. The derivations procedure follows: 



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Step 1: Equation (1.4.7) means that 



,<„...„ = N<-(fx|] 



where 



Nf.=(|..-)/,.(v-x|)/..(|x|)/, 



(1.4.8) 



(1.4.9) 



Step 2: Taking into account that the generating surface Xj performs its relative motion as a rigid body (Xj is 
not changed during such a motion), we may represent in coordinate system ^2 the normal to surface X^ as 



dt') dtn 



99 d(p 



OT*9 OT? 



n(2)= £S.xfS. /„+^x^k + P^x-^ 



50 du 



at') dt'y 



du 99 



(1.4.10) 



(9ri 9rA 
Step 3: Equation gj = yields Np^ = 0, since l'^'^'^}*^ (see eq. (1.4.8)). Respectively, we obtain 

fi^p^ = 0. This means that equation (1.4.10) also yields a normal to surface 2^ that is equal to zero. 
Step 4: We may easily verify that the surface L^ singularity equation can be represented in terms of 2^: 



g2(u,9,(t>) = 



fu 



fe 



(9t£f (dt£\(d^\ (9^ 
\9u) y9u) \99) Vdu 



dt2 
\J9 



~9^ 



dt2 

l9 



u 




y9u) 


{99} 


K^dO ) 





= 



(1.4.11) 



However, equation (1.4.7) is much simpler and therefore preferable. 

We illustrate the phenomenon of undercutting with the generation of a spur involute gear by a rack-cutter 
(figs, 1.4.1 to 1.4.3). The rack-cutter (fig. 1.4.1) consists of a straight line 1 that generates the involute curve 
of the gear, a straight line 2 that generates the dedendum circle of the gear, and the rack fillet that generates the 
gear fillet. 




Figure 1 .4.1 .—Profiles of rack-cutter tooth. 



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Position / 



Position // 




Figure 1 ,4,2. — Detemiination of limiting setting of a 
rack-cutter. 




Figure 1 .4.3. — Generation of involute curve by rack-cutter, (a) Family of shapes of rack-cutter and 
tooth profiles of generated gear with parameter a less than limiting one. (b) Gear involute shape 
undercut by fillet of rack-cutter. 



Figure 1.4.2 shows the generation of a spur gear by the rack-cutter. The rack-cutter setting is determined by 
parameter a. The two rack-cutter positions are designated I and II. At position II, the regular point F of the 
rack-cutter's profile generates a5/ngM/cir point Lof the involute profile; point Lis the intersection of the involute 
profile with the base circle. The appearance of a singular point on the generated profile heralds an undercutting. 
The magnitude of parameter a shown in the figure is the limiting one. 

Figure 1 .4.3(a) shows the family of shapes of the rack-cutter and the tooth profiles of the generated gear when 
parameter a is less than the limiting one. The involute part of the gear tooth profile is free from a singular point, 
the gear fillet and the involute curve are in tangency, and the gear tooth is not undercut. 



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Figure 1 .4.3(b) shows that the fillet of the rack-cutter has undercut the gear involute shape: the gear fillet and 
involute shape are no more in tangency but intersect each other. The undercutting occurred as a result of the 
magnitude of parameter setting a being too large. 

Examples of the application of equation (1.4.7) in order to avoid undercutting are presented in Litvin (1989 
and 1994). 



1.5 Direct Relations Between Curvatures of Meshing Surfaces: 
Instantaneous Contact Ellipse 

The solutions to these related problems are based on the application of the proposed kinematic relations 
discussed in section 1.3. 

Direct relations between the curvatures of meshing gears are necessary for the local synthesis of gear tooth 
surfaces, the determination of an instantaneous contact ellipse, and other problems. The main difficulty in 
solving such problems is that the generated surface L^ is represented by three related parameters, therefore 
making the determination of the curvatures of Zj a complex problem. The approach proposed by Litvin ( 1 968, 
1969, 1994) enables one to overcome this difficulty because the curvatures of 2^ are expressed in terms of the 
curvatures of generating surface E^ and the parameters of motion. Using this approach makes it possible to 
determine the direct relations between the principal curvatures and the directions of Xj and 2^. An extension 
of this approach has enabled one to detemine the relations between normal curvatures of surfaces Xj and 2^ and 
the torsions (Litvin, Chen, and Chen, 1995). The solution is based on the application of equations (1.3.1) and 
(1.3.2) and on the differentiated equation of meshing. 

The developed equations allow one to determine the relationship between the curvatures of surfaces in line 
contact and in point contact. 

The possibility of the interference of surfaces in point contact may be investigated as follows. Consider that 
two gear tooth surfaces designated 2 and 2 are in point tangency at point Af. The principal curvatures and 
directions of 2 and 2 are known. The interrerence of surfaces in the neighborhood of point M will not occur 
if the relative normal curvature k^"^ does not change its sign in any direction in the tangent plane. Here, 



y{r) -yip) _As) 



(1.5.1) 



We are reminded that the normal curvature of a surface can be represented in terms of the principal curvatures 
in Euler's equation. 




Figure 1 .5.1 .—Instantaneous contact ellipse. 



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The approach presented in Litvin ( 1 994) and in Litvin, Chen, and Chen ( 1 995) enables one to determine the 
dimensions of the contact ellipse and its orientation by knowing the principal curvatures and directions of the 
contacting surfaces and the elastic deformation of the tooth surfaces. An alternative approach is based on the 
application of the surface normal curvatures, their torsions, and the tooth elastic approach (Litvin, Chen, and 
Chen, 1995). 

Figure 1.5.1 shows the instantaneous contact ellipse and its center of symmetry, which coincides with the 
point of tangency Af ; the unit vectors ejP^ and ej^> of the respective principal directions; the major and minor 
axes of the contact ellipse 2a and 2b; angle q^^\ which determines the orientation of the contact ellipse with 
respect to the unit vector e}P>; angle a, which is formed by unit vectors ejP^ and e^\ 

1.6 Sufficient Conditions for Existence of Envelope to Family of 
Surfaces 

Sufficient Conditions for Existence of Envelope to Family of Surfaces Represented in Parametric 
Form 

The sufficient conditions for the existence of an envelope to a family of surfaces guarantee that the envelope 
indeed exists, that it be in tangency with the surfaces of the family, and that it be a regular surface. These 
conditions are represented by the following theorem proposed by Zalgaller (1975) and modified by Litvin 
(1968, 1994) for application in the theory of gearing. 

Theorem. Given a regular generating surface Z^ represented in Sy by 



rl(M,0)GC^ 



2 ^i ^ ^i 



L 

du^ de 



^0, (u,9)eE 



The family E^ of surfaces I^ generated in ^2 is represented by r2(w, 0, <^), a < < b. 
Suppose that at a point M (wq, 0q, 0q), the following conditions are observed: 



(t^tjt'^'-"'*'"' 



feC 



(1.6.1) 



(1.6.2) 



or 



(t"!)-™'^'-^-*'" 



(1.6.3) 



fu+fe*0 



(1.6.4) 



«l(«.0.0) = 



fu fe f(p 



/O 



(1.6.5) 



Then, the envelope to the family of surfaces E^ exists in the neighborhood of point M and may be represented 
by 



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Figure 1 .6.1 .—Contact lines on tooth surface. 

r2{uA(t>X /(M,0,0) = O 
The contact lines of 2^ and I^ are represented as 

r2 («, 9, 0), f(u, 0, 0) = 0, <p = Constant 
ti (m, 0), fiu, 0, 0) = a = Constant 



(1.6.6) 

(1.6.7) 
(1.6.8) 



Figure 1.6.1 shows contact lines on surface Xj determined by equations (1.6.8) while constant values of 
<^= (<^l), (^^V..) were taken. 
The tangent to the contact line is represented in 52 and 5j by 



and 






T - *1 f *1 f 



(1.6.9) 



(1.6.10) 



The surface of action is the family of contact lines in the fixed coordinate system 5^ that is rigidly connected 
to the frame. The surface of action is represented by 



r^=r^(M,0,0), /(M,0,0) = O 



(1.6.11) 



where 



10 



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The 4x4 matrix M^ describes the coordinate transformation in transition from 5, to 5^ 



(1.6.12) 



Sufficient Conditions for Existence of Envelope to Family of Surfaces Represented in Implicit Form 

The family of generating surfaces in 52 is 

GUy,z,0) = O, Ged, (G^)^ +(Gp^ +(0,)^ t^O (1.6.13) 

{x,y,z)eA, a<(P<b 

The theorem of sufficient conditions for the envelope existence proposed by Zalgaller (1975) states that at point 
A^ (wq, ^o* ^o)' ^^^ following requirements are observed: 

0{XQ. yo> 20' 00 ) = 0, G^ = 0, G^ ^ 0, 



A = 



D(G,G^) 


+ 


DiCG^) 


+ 


DiG,G^) 


Dix,y) 


D{x,z) 


D(y,z) 



^0 



Thus, the envelope exists locally in the neighborhood of point M and is a regular surface: 

G(^,y,z,0) = a G^(jc,y,z,0) = O 



(1.6.14) 



(1.6.15) 



The surface of action for the case just discussed can be represented by using equations similar to ( 1 .6. 1 1 ) and 
(1.6.12). 



1.7 Envelope Ej to Contact Lines on Generated Surface Xj and Edge of 
Regression 

Sufficient Conditions for Existence of Envelope Ej to Contact Lines 

Singular points on generated surface L^ i"ay form a curve that is the envelope (designated Ej) to the family 
of contact lines (characteristics) on I^. Envelope Ej is simultaneously the "edge of regression" of 2^, which 
means that envelope Ej is simultaneously the common line of two branches of I^ determined by the same 
equation. If the conditions necessary for the existence of Ej as an envelope are not satisfied, singular points on 
generated surface I2 just form an edge of regression. 

Sufficient conditions for the existence of £3 to a family of contact lines on generated surface Xj represented 
by an implicit function were determined by Favard (1957). In this work, it was proven that envelope E2 is also 
the edge of regression. 

Our goal is to present the sufficient conditions for the existence of £3 to a family of contact lines on generated 
surface 2^ that is determined parametrically by three related parameters. In addition, we will show that Ej, if 
it exists, is also the edge of regression. 

Sufficient conditions for the existence of E2 are formulated by the following theorem based on investigations 
conducted by Zalgaller (1975), Zalgaller and Litvin (1977), and Litvin (1975). 
Theorem. A family of generated surfaces Z^ is considered: 



r2(w,^,0) e C^ (a,0) e G, a<(p<b 
The family E^ is generated by surface Xj represented by 



(1.7.1) 



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11 



The following conditions are observed at point M(uq, 9q, 0q): 



^'"■"•^'d^t)- 



„(I2) 



= 



(1.7.2) 



(1.7.3) 



giiu,e,<t>) = 



fu 

du 
dr, ^ (dr, 



de 



fe 



f<p 






= 



(1.7.4) 



fu fe 
8u 89 



*0 



(1.7.5) 



H, = 



du 


de 


v(12) 


fu 


fe 


/0 


8u 


8e 


g^ 



^Q 



(1.7.6) 



Thus, the envelope £3 exists locally at point M(wq, 0q, 0q) and is within the neighborhood of A/. Envelope 
E2 is a regular curve and is determined by 



r2=r2(w,0,^), /(w,^,0) = a g(M,e,0) = O 



(1.7.7) 



The tangent to E2 is collinear to tangent T2 to the contact line at point M of the tangency of E2 and T2. Envelope 
E2 does not exist if at least one of the inequalities ((1.7.5) and (1 .7,6)) is not observed. 

The above theorem was applied by F.L. Litvin, A, Egelja, M. De Donno, A. Peng, and A. Wang to determine 
the envelopes to the contact lines on the surfaces of various spatial gear drives. 

Structure of Curve L on Generated Surface 2^ Near Envelope E2 

We consider curve L on surface I^ that starts at point M of envelope E2 to the contact lines. Since A/ is a 
singular point of surface I^, the velocity yp-^ in any direction that differs from the tangent to E2 is equal to zero. 
Therefore, we may expect that M is the point of regression. A detailed investigation of the structure of curve 
L requires that the Taylor series be applied to prove that M is the point of regression and that envelope E2 is 
simultaneously the edge of regression. 

Example 1.7. 1 . The generation of a helical involute gear by a rack-cutter is considered. The approach discussed 
above is applied to determine the envelope E2 to the contact lines on the generated screw surface ZJj and the edge 
of regression. 

Step 1: We apply coordinate systems Sj, 52, and 5^that are rigidly connected to the rack-cutter, the gear, and 
the frame, respectively (fig. 1.7.1). 



12 



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step 2: The generating surface is plane Xj (fig. 1.7.2). The position vector OjM of point Af of the generating 
plane is 



OlM^OiA-^OiB 



where \OiA\ = 6 and p^ = u . 




Figure 1 .7.1 .—Coordinate systems applied for gener- 
ation of screw involute surface. 



(1.7.8) 




Figure 1. 7.2.— Generating plane: |0i>*|= and |OiB| = u. 



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13 



Then, we obtain the following equation for the generating plane I^: 



ri(w,0) = 



9 cos a f 

Ssina^ + wcosAp 
MsinAp 

1 



(1.7.9) 



The normal Nj to Xj is 






-sina^sin Ap 

cos at sin Ap 

-cosa^cosAp 



(1.7.10) 



Here, a, is the profile angle of the rack-cutter in the transverse section, and Ap is the lead angle on the pitch 
cylinder of radius r of the helical gear. 



Step 3: To derive the equation of meshing, we use the scalar product 

Ni.v</2>=Ni-(v^/^-vf>) = /(M,0,0) = O 

Here (fig. 1.7.1) 



vr^ = -pj 



„(2)- 



v5^^ = kiXri+Oi02Xki 



(1.7.11) 



(1.7.12) 



(1.7.13) 



where kj is the unit vector of the Zj-axis. While deriving equations (1.7.12) and (1.7.13), we have taken 
CO- 1 rad/sec. 
After transformations, we obtain the equation of meshing 



f(u,6,(l>) = Mcos Ap sina^ + - rp0 = 



(1.7.14) 



Step 4: The generated surface 1^, which is the envelope to the family of generating surfaces Xj, is represented 
in coordinate system Sj by 

r2(w,e,0) = M2i(0)ri(M,0) (1.7.15) 

f{uA(t» = (1.7.16) 

where M21 is the matrix for the coordinate transformation from 5j to 52. 
Equations (1.7.15) and (1.7.16) parametrically represent Z^ by three related parameters as 

jC2(M,0,0) = 0cosa^cos0-sin0f0sincif;+MCOsApj + rp(cos0 + 0sin0) (1.7.17) 

}'2(M,0,0) = 6cosa^sin^ + cos0(0sina^ + MCosAp) + rp(sin0-0cos0) (1.7.18) 

Z2(M) = «sinAp (1.7.19) 

/(M,0,^) = wcosApSina; + 0-rp0 = O (1.7.20) 



14 



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Surface branches - 



Contact lines 




Figure 1 ,7.3.— Involute helical gear, (a) Contact lines and envelope to contact lines, (b) Surface 
branches. 



Step 5: Equation (1.7.4) of singularities yields 

^1 (m, 0, 0) = -u cos Xp cos at + rp<j> cos a^ + r^ sin a^ = 



(1.7.21) 



Step 6: The conditions for the existence of envelope Ej to contact lines on L^ formulated by the above theorem 
are satisfied in the case discussed; particularly, there are observed inequalities (1.7.5) and (1.7.6). Therefore, 
envelope Ej indeed exists and is determined by 



r2 (w, e, 0), /(M, 0, <p) = 0, ^1 (w, 0, 0) = 



(1.7.22) 



Equations (1.7. 17) to (1,7.22) yield envelope E2, the helix on the base cylinder of radius r^, and the lines of 
contact, tangents to the helix (fig. 1.7.3(a)) that is represented by 



X2=r^cos(a;+0) 

yi ='fcsin(a^+0) 

Z2 =/7(0 + tana,) 

where r^ = r^ cos a^ the screw parameter p-r tan A . 
Two branches of the generated surface are shown in figure 1.7.3(b). 



(1.7.23) 
(1.7.24) 
(1.7,25) 



Step 7: The generation of a spur gear may be considered a particular case of the generation of a helical gear by 
taking A^ = 90°. In such a case, envelope Ej to the contact lines does not exist since inequality (1.7.5) is not 
observed. Only the edge of regression exists, as shown in figure 1.7.4. 



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15 



Branches 
ofSn 



cylinder -^^ X j/yVpS. / regression 




Figure 1,7.4.— Edge of regression of spur involute surface. 



1.8 Necessary and Sufficient Conditions for Existence of Envelope Ej to 
Contact Lines on Generating Surface I^ 

Necessary Condition 

Contact lines on generating surface Xj may also have an envelope Ep This envelope divides the generating 
surface into two parts: A, which is covered with contact lines, and B, which is empty of contact lines (fig. L8 . 1 ). 
Envelope Ej was discovered by Litvin (1975) and the necessary conditions for the existence of Ej were 
formulated. In addition, sufficient conditions for the existence of Ej are formulated in this book. 

The family of contact lines on Zj is represented in 5j by the expressions 

ri(M,0)eC^ -^x-^^^O, /(m,0,0) = O, (w,0)eG, a<(^<b (1.8.1) 

du ad 

The necessary condition for the existence of envelope Ej is/^ = 0, The proof is based on the following 
considerations (Litvin, 1975): 

(1) Vector 5ri of displacement along the tangent to a contact line may be represented by 

S^ =^Sux^Se, f,5u-\-fe59^0 (1.8.2) 

au oB 

(2) Vector drj of displacement along the tangent to envelope Ej may be represented by 

dr, =^dw + ^d0, /„dw + /0de + /^d0 = O (1.8.3) 

du aU 

(3) Vectors 5r^ and dr^ must be collinear if envelope Ej exists. This requirement is satisfied if /^d^ = 0, 
which yields/^ = (d0 ;t since is a varied parameter of motion). 

Sufficient Conditions 

The generating surface X| is represented as 

r,(«,9)€C2, ^x%^0, (M,e)eG, a<(t><b (1.8,4) 

du ad 

16 NASA RP-1 406 




Figure 1.8.1. — Envelope to contact lines. 



The following conditions are observed at a point {Ug,dg,(pg) designated M: 

nu,e,(t>) = o 



^0 



(1.8.5) 
(1.8.6) 
(1.8.7) 

(1.8.8) 
(1.8.9) 



fu fe 
fipu f(pe 

Then, envelope E, exists, is a regular curve, and is determined by 

r,(«,0), /(M,0,0) = O, f^(u,d,<l>) = 

The proof of the theorem of sufficient conditions is based on the following procedure: 

Step 1: Consider the system of equations (1.8.5) and (1.8.6) and apply the theorem of implicit equation system 
existence. We can solve these equations in the neighborhood of point (Up, 0^,0^) by functions {«(0), diip)] e C' 
since inequality (1.8.7) is observed. Then, we may determine a curve on surface E^ to be 



R(0) = r,(M(^),0(^)) 



(1.8.10) 



Step 2: The tangent to curve R(0) is determined as 

d^ du d(t> de dip 

Step 3: We may determine the derivatives du/d0 and d9/d(j) as follows: differentiating equations ( 1 .8.5) and 
(1.8.6), we obtain 



, dM . d0 _ , 
^"d0"'^^d^""^^ 

- dM f de _ . 

Solving equations (1.8.12) and (1.8.13), we obtain dM/d0and d0/d0. 
Equations (1.8.5), (1.8,10), and (1.8.11) to (1.8.13) yield 

dR fdxb 



(1.8.12) 
(1.8.13) 



fu fe 
fipu fipe 






(1.8.14) 



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17 




Figure 1 .8.2. — Envelope to contact lines on womi 
surface. 



where 






(1.8.15) 



is the equation of the tangent to the contact line. 

Equations (1.8.14) and (1.8.15) confirm that R(0) is a regular curve (due to inequalities (1.8.7) and (1.8.8) 
and Tj ^ 0); the tangent dR/d(p is collinear to tangent Tj to the contact line, and R(0)4s indeed the envelope to 
the family of contact lines on Xj. 

Note 1: Equation (1.8.5) represents a family of planar curves in the space of parameters (m,0). Using an 
approach similar to that just discussed, it is easy to verify that/^ = QJ^^^O, and inequalities (1 .8.7) and ( 1 .8.8) 
are the sufficient conditions for the existence of an envelope to the family of lines of contact between surfaces 
Ey and I^ represented in the space (m,0). 

Note 2: The envelope E J onXj is a regular curve formed by regular points of Xj. A point of Ej may generate 
a singular point on I^ if and only if the relative velocity v{^^^ is collinear to the tangent 3Rld(p to the envelope 
Ej on Xj. This conclusion can be drawn from equation (1.4.9), which is represented after transformation as 



N' 



(2)_^{12) 



= V 



{^^-l^-JHt^t^'-i'-'^txt^ <>-^' 



The normal N/^^ to 2^ may become equal to zero if/^^ and v j'^^ is collinear to Tj (or T2), which is tangent 
to the contact line. 

Figure 1.8.2 shows contact lines and the envelope to such lines on the generating surface that is the worm 
thread surface. The generated surface is the worm-gear tooth surface. Figure 1,8.3 shows the contact lines and 
their envelope in the space of the worm surface parameters. 



18 



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/^ Contact lines 




Envelope 



Figure 1 .8.3.— Contact lines in space of surface 
parameters. 



1.9 Axes of Meshing 

Initial Considerations 

The concept of the axes of meshing and their derivation was first presented by Litvin in 1955 and was then 
pubhshed in the works (Litvin, 1968, 1989). The revised and complemented concept is now presented in this 
book. 

We consider that a gear drive transforms rotations with angular velocities 00^ ^ ^ and (O^^^ between crossed axes. 
The angular velocities co^ * ^ and (jp^ lie in parallel planes and form a crossing angle y, with the shortest distance 
between (0^*^ and co^^) being E (fig. 1.9.1). 

The relative motion of gear 1 with respect to gear 2 is represented by vector ©^ ^^) = ©^ * > - co^^) ^^^ ^y vector 
moment mf-co^^^j = Of02 x(-g}^^A 

A point M is the current point of tangency of gear tooth surfaces Xj and Zj if the following equation (the 
equation of meshing) is observed: 



nv 



(12) 



= n.{(a)(»2)xr) + [o;^x(-a>(2))|^0 



(1.9.1) 



where r is the position vector of A/, and n is the unit normal to surface Z,. The normal N to surface Xj instead 
of unit normal n may be applied in equation ( 1 .9. 1 ). Henceforth, we will consider that vectors in equation ( 1 .9. 1 ) 
and those derived below are represented in the fixed coordinate system 5^ (fig. 1.9. 1). 

It is known from kinematics that the same relative motion will be provided if vectors (0^^^ and co^^^ are 
substituted by vectors 00^^ and ©^^^^ and if the following conditions are observed: 

(1) Vectors ©^^^ and ©^''> lie in planes 77^^^ and n^^^\ which are parallel to ©^*^ and (iP\ 

(2) Vectors ©^^ and ©^'^^ are correlated and satisfy these equations: 



©<^>-©W=©(i2) 



0^0^'^ x©(^> +0;^^^'^ (-©('')) = o;^x(-©(2)) 



(1.9.2) 
(1.9.3) 



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19 




^f—i 



n 



m 



7 


L:^y 


ow 






Y 


N^rfo / 




0?, 


^ J 7 






Figure 1 .9.1 .—Derivation of axes of meshing. 



General Concept of Axes of Meshing 

A manifold of couples of vectors (O^^ and (0^'^> satisfy equations (1.9.2) and (1.9.3). We will consider a 
submanifold of vectors (0^^ and co^^^^ that not only satisfy equations (1.9.2) and (1.9.3) but also satisfy the 
requirement that the common surface normal N (or the unit normal n) intersect the lines of action L^^ and L^^^^ 
ofco^^ and co^^'Ufig. 1.9.2). 

If the normal (unit normal) at point M intersects at least one of the couple of lines L^^ and L^^^\ say L^^, it 
is easy to prove that two equations of meshing are satisfied and that the normal intersects the other line, L^^^\ 
The proof is based on these considerations: 



(1) An equation of meshing similar to equation (1.9.1) can be represented as 

nv<^-")=n(v(^>-v<")) = 

(2) If the normal intersects L^'^, we can represent v<') by 



yO)=^0)^pU) 



(1.9.4) 



(1.9.5) 



where pf-^ is a position ve ctor dra wn to M from any point on the line of action L^^. Not losing the generality, 
p<') can be represented as P^'^M, where P*-^ is the point of intersection of L^*^ and the extended unit normal n. 



20 



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of meshing 




^— Axis of meshing 

Figure 1 .9.2.— Intersection of axes of meshing by the 
normal to the contacting surfaces. 



Thus, we have 



iuA(t» = 



(3) Considering the scalar product n • v^^, we obtain 

(4) Equations (1.9.4) and (1.9,7) yield 



.v(''>=0 



(1.8,6) 



(1.9.7) 



(1.9.8) 



The foregoing discussions mean that (a) if the surface normal at point M (the candidate for the point of 
tangency of surfaces) intersects at least one line of the couple of lines L^'^ (/ = /,//), it intersects the other line 
as well; (b) two equations of meshing, (1.9.7) and (1,9.8), are satisfied simultaneously; and (c) point M is the 
point of tangency of the surfaces if at least one equation of meshing of the couple ( 1 .9.7) and ( 1 .9.8) is satisfied. 

We call lines L^^ and L^^^ the axes of meshing. However, we emphasize that the couple of lines L^^ and L^'^^ 
must satisfy not only equations (1 .9,2) and (1.9.3) but also the requirement that L^^ and L^''^ be intersected by 
the surface normal. Lines L^^ and L^^^^ that satisfy requirements (a) to (c) are called the axes of meshing. 



NASA RP-1 406 



21 



Correlation Between Parameters of Axes of Meshing 

Our goal is to prove that the parameters of the axes of meshing are correlated, that they depend on the position 
vector r and the surface normal N, and that the solution for determining the parameters of L^^ and L^^^ is not 
unique. 

The determination of the sought-for parameters of the axes of meshing is based on the following procedure: 

(1) The axes of meshing lie in planes that are perpendicular to the x^-axis (fig. 1.9.1), and we designate the 
sought-for parameters ^\ KS^^ (i = /, //). The algebraic parameter X^^ determines the location of O^^^ on axis 
Xj-; parameter /^'^ is determined as 



r'^=% (1.9.9) 



r(0-.^' 



^'^ 



(2) The requirement that the surface unit normal pass through the axes of meshing is presented in the 
following equations (Litvin, 1968): 



^ - = ' ^=^- ' {i = Lin (1.9.10) 



riy 



Here (fig. 1.9.2) 



<9^f(') = (x^'>, y('), Z^') = ^^')y^'>)(/ = /, //); OfM = r = (X, y, z)\ n = (n^, n^, n, ) 

where f^'^ is the point of intersection of the normal with the axis of meshing. 
After eliminating Y^^ in equations (1.9.10), we obtain 

X^'^K^'^fiy - X^% + K^'^yn^ -xrty)-^ xn, - zn^ = (/ = /, //) (1 .9.1 1) 

Equation (1.9.11) yields 

(;t(/)^(/) _ x^^KUny^ _(x(^) - X<"))n, +(a:(^> - K^'%yn, -xn^) = . (1.9.12) 

(3) Additional relations between parameters X^'\ ^'^ (/ = /,//) can be obtained by using equations ( 1 .9.2) and 
(1.9.3). These equations yield the following four dependent scalar equations in the unknowns co^^ and fi)j^'^: 



(0 



(^)-a?^'^)=-m2isiny (1.9,13) 



K^^^co^J^ - K^^^^mf^ = 1 - /n2i cos y (1 .9.14) 

X^'^K^'^'^ - X^^^^K^"W^"^ = £m2i cosy (1.9.15) 

X^^^co^J^ - X^^^Wy^^^ = Em2x sin y (L9.16) 

While deriving these equations, we assign the value of CO^*^ to be 1 rad/sec and dP^ to be m2j rad/sec, where 
m2i is the gear ratio. The system of equations (1.9.13) to (1.9.16) in the unknowns O)^^ and Q)^^^^ may exist if 
the rank of matrix 



22 NASARP-1406 



1 -1 -m2i siny 

xOkU) .xUD^Un £^21 cosy 



(1.9.17) 



IS two. 



Taking into consideration that in matrix ( 1 .9. 1 7) the four respective determinants of the third order are equal 
to zero, we obtain after transformations the relations 



(,) ^ £m2i(cosy-Ar^^^^siny) 
1 - ^21 cos y + K^"^m2 1 sin y 

_ £>n2i(cosy-X^in y) 
1 - m2i cos y + K"-' ^m2\ sin y 



(1.9.18) 



(1.9.19) 



(4) By analyzing the system of equations (1.9.12), (1.9.18), and (1.9.19), we can conclude the following: 

( 1) The parameters X^'^ /^'^ (/ = /, //) of the axes of meshing depend on the coordinates (jcj) of the contact 
point of the surfaces and on the components of the surface normal (unit normal). 

(2) Three equations relate four parameters of the axes of meshing. The solution for the parameters of the 
axes of meshing is not unique, even when the point of tangency and the common normal to the contacting 
surfaces are considered known. This means that for any instant of meshing, there is a manifold of the axes of 
meshing. However, there are two particular cases of meshing when only a couple of the axes of meshing, but 
not a manifold of such axes, exist and the parameters of the axes of meshing do not depend on the contact point 
and the contact normal: case 1 , where the rotation is performed between crossed axes and the surface of one of 
the gears is the helicoid; case 2, where a helicoid is generated by a peripheral milling (grinding) tool whose 
surface is a surface of revolution. 

Case 1 has been applied in the analysis of the meshing of worm-gear drives with cylindrical worms, helicon 
drives, face-gear drives with crossed axes, and some types of spiroid gear drives. Case 2 has been applied in 
the generation of worms and helical gears by a peripheral cutting (grinding) disk. 

Case 1 of axes of meshing. — ^The surface of one of the mating gears is a helicoid. To derive the parameters 
of the axes of meshing, we apply equations (1.9.12) and (1.9.13) to (1.9.16) and require that the sought-for 
parameters be independent with respect to the point of tangency of the mating surfaces. 

In the case of a helicoid, we have the following relation (Litvin, 1968): 



yn^ - xriy = pn^ 



(1.9.20) 



where p is the screw parameter of the helicoid. 
Equations (1.9.20) and (1.9.12) yield 



(;^(/)^(/) . x^m^imy^ _ [(;^(/) _ ^(/)^) _(;^(//) _ ^(//)^)]„^ = o (1.9.21) 

Equation ( 1 .9.2 1 ) shows that the parameters of the axes of meshing do not depend on components of the surface 
normal if these relations are observed: 



X^I)fiU)^X^"^K^"^ 



(1.9.22) 



NASA RP-1406 



23 



An 



(/)„_ y(//)_ji'(/'), 



,X^'>-K''>p = X""-K"'>p 



(1.9.23) 



Further derivations of X<'\ ^'^ (/ = /, //) are based on the application of equations ( 1 .9.22) and ( 1 .9.23) and 
the system of equations (1.9.13) to (1.9.16). The procedure for deriving the parameters follows: 



Step 1: Considering equations (1.9.22), (1.9.13), and (1.9.15), we obtain 

Step 2: Considering equations (1.9.14) and (1.9.16), we obtain after transformations 

(X^'^ - pK^'Yy'^ - (X^"^ - pK^"^\of^ = Em2i sin y - Kl - mji cosy) 

Equations (1.9.25), (1.9.23), and (1.9.13) yield 

£'m2isiny-/>(l-m2i) , • „ n /• i n\ 

— ii—jj( — i-^ — ^J- + /W2ismy = (1 = 1.11) 



X^'^-pK 



0) 



(1.9.24) 



(1.9.25) 



(1.9.26) 



Then, using equations (1.9.26) and (1.9.24), we obtain 

■ Ifc^of _ (l _ l-^2lC0sy V(,) ^ £coty ^ Q 
V / [p m2ismy ) p 



(1.9.27) 



Step 3: The final equations for the determination of parameters A<'\ K^'^ of the couple of the axes of meshing 
are 

r . ,1 nO.5 

^ E l-W2icosy'^ 



K^'^=^ 



^p m2isiny 



1 
+ - 
2 



E l-m2icosy | 4Ecoiy 
p m2isiny J p 



(1,9.28) 



(/)__£coty 






E l-m2iCosy 
p m2isiny 



4£coty 



0.5 



(1.9.29) 



(1.9,30) 



(//)_ EcoiY 



(1.9.31) 



In the case of an orthogonal gear drive, we have Y= 7c/2. To determine the expression for X^^^ = 0/0, we use 
the Lopithal rule (Korn and Kom, 1968). Then, we obtain the following equations for the axes of meshing 
parameters: 



-(/)_.^ 



1 



P rn2x 



(1.9.32) 



x(^> = o 



(1,9.33) 



24 



NASA RP-1406 



(a) 




// 







i 




/ 

/ 










7 






'-r 


^ 


^ 


/ 




» 




J 


1 









// 




Axis of meshing / 



Axis of meshing // 



Figure 1 .9.3.— Axes of meshing of orthogonal womi-gear 
drive, (a) In three-dimensional space, (b) In orthogonal 
projections. 



NASA RP-1406 



25 



EcoXyiankp || 




meshing / 



Axis of 
meshing// 

Figure 1 .9.4.— Axes of meshing of nonorthogonal 
worm-gear drive with right-hand worm, (a) In three- 
dimensional space, (b) In orthogonal projections. 



26 



NASA RP-1406 




Figure 1 .9.5,— Generation of worm by peripheral tool. 




Figure 1 .9.6.— Axes of meshing in case of helicoid 
generation. 



NASARP-1406 



27 



K^^^=0 (1.9.34) 

X^m^JE—E^] (1.9,35) 

The axes of meshing for a worm-gear drive with a cylindrical worm are shown in figures 1.9.3 and 1.9.4. 

For a face-gear drive with crossed axes and with a pinion as a spur gear, we have to take /? = «». 

In this case, all contact normals are perpendicular to the pinion axis, and one of the axes of meshing lies in 
the infinity. 

Case 2 of axes of meshing, — We will consider the generation of a helicoid by a cutting or grinding disk. The 
installment of the tool is shown in figure 1.9.5. Coordinate system 5^ is rigidly connected to the frame. The 
helicoid in the process of generation performs a screw motion about the z^-axis with the screw parameter^. We 
may neglect the tool rotation since it is provided to obtain the desired cutting velocity and does not affect the 
process of generation. Therefore, we may consider that systems S^ and S^ are rigidly connected during the 
generation process. The crossing angle y^ between the z^,- and z^-axes is usually equal to the lead angle X^ on 
the helicoid pitch cylinder. The shortest distance is £^. 

There are two axes of meshing in this case: /- /coincides with the tool axis; // - // lies in the plane that is 
perpendicular to the shortest distance E^ (fig. 1 .9.6). The shortest distance between the helicoid axis and the axes 
of meshing// - //is 

fl = 4^')=;7coty^ (1.9.36) 

1.10 Two-Parameter Enveloping 

The motion of the generating surface E^ is determined with two independent parameters designated (0,vO, 
and the family of surfaces Xj is represented in 52 as 

r2(w,0,0,v^) = M2i(0,i^)ri(M,0) (1.10.1) 

The two equations of meshing for the case of two-parametric enveloping are determined as 

Ni-vj'^>=/(w,0,(t),i|/) = O, N,-v|^^=g(M,0,(t),V) = O (1.10.2) 

Here, v/^^ = y iiZ(i>)^ y (v) = Vj^^^'VO represent the sliding velocity when the respective parameter of motion 
(0or y/) is fixed. The subscript 1 in equations (1.10.2) indicates that the respective vectors are represented in 
coordinate system Sy, 

The two-parameter method of enveloping was discussed in Litvin, Krylov, and Erikhov (1975) and Litvin 
and Seol ( 1 996). It can be successfully applied when the tool has a feed motion in the generation processes, such 
as bobbing, shaving, and grinding. We have to emphasize that in reality the generation of surfaces with feed 
motion is a one-parameter enveloping process because the two parameters of motion, and i^, are related by 
the generating function v^0). Using one-parameter enveloping makes it possible to determine the real surface 
2^ and its deviation from the theoretical envelope 2^ and to evaluate the influence of the feed motion (of 
function V^0)), 

A detailed example of two-parameter enveloping is presented in appendix C, 

1.11 Localization of Contact and Simulation of Meshing of Gear Tooth 
Surfaces 

The localization of gear tooth surface contact is achieved when point contact instead of line contact of the 
surfaces is provided. This enables one to reduce the sensitivity of the gear drive to misahgnment and to also 

28 NASARP-1406 



avoid so-called edge contact. The localization can be achieved by the mismatch of gear tooth surfaces. The 
Gleason Works engineers have successfully developed spiral bevel gears and hypoid gears with point contact 
of the surfaces. A localized contact is provided for circular-arc helical gears (Novikov-Wildhaber gears) and 
can be achieved for other types of gear drives by gear tooth surface crowning. 

We consider a great achievement to be the computerized simulation of meshing and of gear tooth surfaces 
in point contact accomplished by applying TCA (Tooth Contact Analysis) computer programs. The simulation 
of the meshing of gear tooth surfaces is based on the conditions of continuous tangency of gear tooth surfaces 
that are represented by the following equations 

rf\u^AA^^j) = ^fHu2.e2.<p2^qj) (1.11.1) 

^fi^lA^<Pu<ij) = nf\u2.e2.(l)2,qj) (1.11.2) 

Equations (1.1 1.1) and (1.1 1.2) indicate that the contacting surfaces have at the current point of tangency 
common position vectors rj^^^ and surface unit normals nP\ (i = 1 ,2). The coincidence of directions of the unit 
normals for both surfaces can be provided by the proper order of cofactors in the cross products (3ry V3m -) 
X (3rj'V30.), (i = 1 ,2). The gear tooth surfaces I^ and I^ are represented in the fixed coordinate systems 5^ where 
the axes of gear rotation are located; (w ., 6-) (i - 1 ,2) are the surface parameters; 0, and 02 are the angles of gear 
rotation; qj (j = 1,2,...) designate the parameters of assembly. 

Equations (1.11.1) and (1,11.2) yield a system of only five independent nonlinear equations (in six 
unknowns) since |ny!^| = |n^^^ =1, These equations are represented as 

A(M,,0,,0,,M2>e2»02) = O, fkeC\ (fc^lS) (1.11.3) 

One of the unknowns, say 0( , may be chosen as the input. Henceforth, we assume that equations ( 1 . 1 1 .3) are 
satisfied at a point 

p' = (uf,e?,<i,?A>el4) (1.11.4) 

and the Jacobian system at /^differs from zero. Thus, 

^^=^fllMtM^^O (1.11,5) 

From the theorem of the existence of the implicit function system, it follows that equations (1.1 1,3) can be 
solved in the neighborhood of /^ by functions 

h(^i)>ei(0i)»"2(0i)^e2(0i>02(^i)]eC* (1.11.6) 

By using equations (1.11.1) and (1.11.2) and functions (1.11,6), we can determine the paths of contact on 
surfaces Xj and L^ and the transmission function 02(0i)- The gear misalignment is simulated by the variation 
of assembly parameters qj that will cause the shift in the paths of contact and the deviations of (pj (0i) from the 
transmission function of an aligned gear drive. Note that a unique solution of equations (1.1 1.3) by functions 
( 1 . 1 1 .6) exists only for the case of meshing by a point contact of surfaces. The Jacobian A5 becomes equal to 
zero when the surfaces are in line contact. 

This method of simulating the meshing of misaligned gear drives was proposed by Litvin and Guo (1962). 
We must credit The Gleason Works researchers who developed and applied in industry the TCA computer 
programs for hypoid gear and spiral bevel gear drives. Similar programs were developed later by Litvin and 
Gutman(1981). 

The numerical solution of nonlinear equations (1.1 1.3) is an iterative process based on the application of 
computer programs (Dongarra et al., 1979 and Mord, 1980). 



NASARP-1406 29 



1.12 Equation of Meshing for Surfaces in Point Contact 

The computerized simulation of the meshing of gear tooth surfaces described in section 1.11 does not require 
knowledge of the equation of meshing. However, it is possible to prove that equation (1.2.7) can also be 
extended to apply in the case of surfaces in point contact. 

Consider an aligned gear drive when the assembly parameters are observed. The differentiation of equation 
(1.1 1.1) yields 



It is easy to verify that the derivatives (3rj'V3M^), (3rj'V30.) (/ = 1,2) lie in the common tangent plane for 
surfaces in tangency. Using the scalar product of the common surface normal Ny^'^ with both sides of equa- 
tion (1.12.1), we obtain 



^d^_^d^ 



nJ^=0 (1.12.2) 



Taking into account that (3ri'V30,) (d^/d/) is the velocity v^{? of the surface point (in transfer motion with 
the surface), we obtain 

NSJ>-(vi:^-v{,2)) = Nj)-vfU0 (1.12.3) 

This is the proof that the equation of meshing can also be applied for the case of the point contact of surfaces. 
Similar derivations performed for a misaligned gear drive yield 



Nf 



^d0i--^d02-^cl^,- 



= (1,12.4) 



which allows us to investigate the influence of gear misalignment. However, this equation can be applied when 
the theoretical line of action (the set of contact points in the fixed coordinate system) is known. The influence 
of the misalignment errors can be determined directly by applying the TCA program. 

1.13 Transition From Surface Line Contact to Point Contact 

Instantaneous line contact of gear tooth surfaces may exist only in ideal gear drives without misalignment 
and manufacturing errors. Such errors cause the gear tooth surfaces to contact each other at a point at every 
instant instead of on aline. The set of contact points on the gear tooth surface forms the contact path. A current 
point of the contact path indicates the location of the center of the instantaneous contact ellipse, (Recall that 
because of the elastic deformation of the teeth, the contact is spread over an elliptical area.) The set of contact 
ellipses represents the bearing contact, which covers only a certain part of the tooth surface instead of the entire 
working tooth surface (in the case of the line contact of an ideal gear drive). 

Our goals are to determine the following: (1) the contact path for a misaligned gear drive and (2) the 
transmission errors caused by misalignment. Such problems are important for those gear drives whose gear 
tooth surfaces are designed as mutually enveloping. Typical examples are a worm-gear drive with a cylindrical 
worm and involute helical gears with parallel axes. 

Figure 1.13. 1(a) shows two neighboring contact linesLj(0) and L2(0+d0) on surf ace X, of an ideal gear drive 
without misalignment. Surface X, is in instantaneous Une contact with Xj; parameter (p is the generalized 
parameter of motion, and point A/ is a current point of contact line 1^(0). The displacement from M to any point 
on L2{<p + d0) can be performed in any direction if it differs from the tangent to line Lj(0) at M. However, in 
the case of a misaligned gear drive with a point contact of surfaces Xj and X^, we have to determine (1) the 
transition point P on the contact line Li(0) (fig. 1 . 13. 1(b)) (the transfer from line contact of the surfaces to point 

30 NASA RP-1 406 



Contact line 



L2(<!> + d<}>) 




^ Path of 
contact 
points 

Figure 1 .1 3.1 .—For derivation of transition point, (a) Representation of two neighboring contact lines, 
(b) Transition from surface point P to P* via K. 



Contact path 




Figure 1 .1 3.2.— Contact path of misalignment of 
worm-gear drive. Change of center distance, 
d£ = 0.5 mm; change of shaft angle, A7 = 5'. 



contact will occur in the neighborhood of P) and (2) the current point P* of the real contact path. The direct 
determination of P* is impossible because theJacobian A5 of the system of equations (1.12.3) of surface 
tangency is equal to zero. Therefore, it becomes necessary to determine an intermediate point K in the 
neighborhood of P (fig. 1.13.1 (b)) where A5 differs from zero. The determination of point K is based on the fact 
that vector PK is collinear to the vector that passes through two neighboring transition points. An equation to 
determine the transition point on a contact line L j (0) was proposed in Litvin ( 1 994) and Lit vin and Hsiao ( 1 993). 
The Jacobian A^ at point K differs from zero, and we can start the procedure of simulating the meshing of two 
surfaces in point contact. 

Figure 1.13.2 shows the shift in the bearing contact in a misaligned worm-gear drive. Transmission errors 
due to misalignment will occur and may cause noise and vibration (see section 1.14). 



NASARP-1406 



31 



1.14 Design and Generation of Gear Drives With Compensated 
Transmission Errors 

Influence of Transmission Errors on Conditions for Transfer of Meshing 

Experimental tests show that the level of noise and vibration depends on the level and shape of transmission 
errors caused by gear misalignment. Henceforth, we will assume that the gear tooth surfaces are mismatched 
and that they contact each other at every instant at a point. This precondition is important when designing low- 
noise gear drives, but it must be complemented with the requirement that one apply the predesigned parabolic 
function of transmission errors, which is represented as 

402(0l) = -«0? (1.14.1) 

It will now be shown that the application of such a function allows one to absorb transmission errors caused 
by gear misalignment, to avoid edge contact, and to improve the conditions for the transfer of meshing. Edge 
contact means curve-to-surface contact that may occur instead of surface-to-surface contact. In such a case, the 
curve is the edge of the gear tooth surface of one of the mating gears that is in mesh with the tooth surface of 
the mating gear. The transfer of meshing means that the continuous transformation of motions by a gear drive 
requires that a pair of teeth in mesh be changed for another pair. 

Figure 1 . 1 4, 1 (a) shows that the transmission function 02(<?i ) for an ideal gear drive is linear and is represented 
as 

02(0i) = ^0l a-14.2) 

where A^j and N2 are the gear tooth numbers. The contact ratio (the number of teeth being in mesh 
simultaneously) may be larger than 1 in an ideal gear drive. In reality, ideal gear drives do not exist because 
alignment errors cause transmission errors that substantially worsen the conditions for the transfer of motion. 
Figure 1 . 1 4. 1 (b) shows the transmission function 02(<?i ) for a misaligned gear drive that is a piecewise nonlinear 
function for each cycle of meshing with worsened conditions for the transfer of meshing. The cycle of meshing 
is determined with the angles of rotation of the driving and driven gear represented as 0| = {InlN^) and 
02 = {2nlN^. The author and his fellow researchers at the University of Illinois investigated crowned involute 
helical gears, double-circular-arc helical gears, and hypoid gears. They found that the function of transmission 
errors A02(0i) for misaligned gear drives usually has the shape shown in figure 1.14.2(a). The linear part of 
A02(0i) is caused by gear misalignment; the nonlinear dashed part of A02(0i) corresponds to the portion of the 
meshing cycle when the edge contact occurs. The second derivative of A02(0i), and therefore the acceleration 
of the driven gear, makes a big jump at the transfer point A of the meshing cycle. 

The author's approach is directed at improving the conditions for the transfer of meshing and is based on the 
application of a predesigned parabolic function (1.14.1) of transmission errors. Such a function is provided by 
the proper modification of gear tooth surfaces or by the stipulation of specific relations between the motions 
of the tool and the generating gear in the generation process. It will be shown next, that the simultaneous action 
of both transmission error functions, the predesigned one and that caused by misalignment (in fig. 1.14.2(a)), 
causes a resulting function of transmission errors that is again a parabolic function having the same slope as the 
initially predesigned parabolic function. The magnitude A02max of the resulting maximal transmission errors 
(caused by the interaction of both functions shown in fig. 1 .14.2(b)) can be. substantially reduced. The level of 
the driven gear accelerations is reduced as well, and an edge contact, as a rule, can be avoided. 

The transmission function for the gear drive, when the predesigned parabolic function of transmission errors 
is provided, is shown in figure 1 . 1 4.3(a). The predesigned parabolic function is shown in figure 1 . 1 4.3(b). It is 
important to recognize that the contact ratio for a misaligned gear drive with rigid teeth is equal to 1 . However, 
the real contact ratio is larger than 1 because of the elastic deformation of the teeth. While investigating the 
correlation between the predesigned function of transmission errors and the elastic deformation of teeth, we 
have to consider the variation in the elastic deformation of the teeth during the meshing process, but not the 
whole value of the elastic deformation. It is assumed that the variation in elastic deformation is comparable to 
the level of compensated transmission errors. 



32 NASARP-1406 





•^<t>1 



(b) 

Figure 1 .1 4.1 .—Transmission functions of ideal and misaligned gear drives, (a) Ideal, 
(b) Misaligned. 




A<J)2 




\ ^ ^<l>2max 



(a) (b) 

Figure 1 .1 4.2.— Function of transmission enters for gears, (a) Existing geometry, (b) Modified geometry. 




A<t>2 




(a) 

Figure 1 .1 4.3.— Transmission function and function of transmission enters for misaligned gear drive, 
(a) 1 , ideal transmission function; 2, transmission function for gears with modified geometry, (b) 3, 
predesigned parabolic function of transmission errors. 



NASARP-1406 



33 



A(|>2 Aijio 



A<|>f = /)4>i 



:^^^ 




A<t>2 






^■'^A<t,J)=-a<t>f 



(a) 




— ^I'l 



(b) 



Figure 1 .1 4.4.— Interaction of parabolic and linear functions, (a) Linear and parabolic functions of 
transmission errors, (b) Resulting function of transmission errors. 

Interaction of Parabolic and Linear Functions of Transmission Errors 

Figure 1 .14.4(a) shows the interaction of two functions: (1) the linear function A02^*^ = b<Pi caused by gear 
misalignment and (2) the predesigned parabolic function A<j)l^^ = -a(j>f provided by the modification of the 
contacting gear tooth surfaces. Our goal is to prove that the linear function A<l>^^\<p^) will be absorbed because 
of the existence of the parabolic function A(p^^ = -a0p To prove it, we consider the resulting function of 
transmission errors to be 



zi02(0i)-M*^(0i) + M^^(0i) = ^0l-#? 



(1.14.3) 



The proof is based on the consideration that equation (1.14.3) represents in a new coordinate system with axes 
(Av^2» V^i) (^^S- 1.14.4(a)) the parabolic function 



A\i/2 =-ci¥\ 



(1.14.4) 



The axes of coordinate systems (Ai/a^, y/^) and (A<^2» ^i) ^^ parallel but their origins are different. The 
coordinate transformation between the coordinate systems above is represented by 

u2 u 



A\i/2=^<t>2--r^ 
Aa 



W.-^^-Ya 



(1.14.5) 



Equations (1.14.3) and (1.14.5), considered simultaneously, yield equation (1.14.4). Thus, the Hnear function 
A^i*^ (0j) is indeed absorbed because of its interaction with the predesigned parabolic function A0^^^ (0j). This 
statement is in agreement with the transformation of equations of second-order curves discussed in the 
mathematics literature (Korn and Korn, 1968). 

The difference between the predesigned parabolic function A02^^^ (0|) and the resulting parabolic function 
^WoiWO ^^ ^^^ location of points (y4*,B*) in comparison with (A,5). Figure 1 . 1 4.4(a) shows that the symmetrical 
location of {A,B) is turned into the asymmetrical location of (A*,fi*). However, the interaction of several 
functions Av^2(V^i)» determined for several neighboring tooth surfaces, provides a synrunetrical parabolic 
function of transmission errors A v^2( V\ ) as shown in figure 1 . 1 4.4(b). (The neighboring tooth surfaces enter into 
mesh in sequence.) The symmetrical shape of function Av^2(V^i) determined for several cycles of meshing can 
be achieved if the parabolic function A^^\(py) is predesigned in the area (fig. 1.14.4(a)) 



0i(5)-02(A)>^ + - 



(1.14.6) 



The requirement (1,14.6), if observed, provides a continuous symmetrical function A\i/2{y/{) for the range of 
the meshing cycle 0j = InlN^ 



34 



NASARP-1406 



Appendix A 
Parallel Transfer of Sliding Vectors 

A sliding vector a is determined by its magnitude |a| and the line of its action Aq-A (fig. A. 1 ), along which 
it can be moved. Examples of a sliding vector are forces and angular velocities. In the last case, the line of action 
Aq'A is the axis of rotation. 

The parallel transfer of a sliding vector means that a can be substituted by a* = a and a vector moment 

m = Rxa (A.I) 

where R is a position vector drawn from point O of the line of action of a* to any point of the line of action of 
a. It is easy to prove that the vector moment m, for instance, can be expressed as (fig. A.l) 

m = a4^xa (A.2) 

Taking into account that 

R = aA^ + AoA* = a4^+Aa (A.3) 

where A is a scalar factor, we obtain 

m = R X a = (oAq + Aa) x a = OAq x a ( A.4) 

Figure A.2 is an example of a shding vector as the angular velocity of rotation (W about the axis A^-A. The 
axis of rotation does not pass through the origin O of the considered coordinate system S{x,y,z). The velocity 
of point M in rotation about Aq-A can be determined using the following procedure: 

Step 1: We substitute vector (O that passes through Aq by a parallel and equal vector (tf^ that passes through 
and the vector moment 

01 = 04^X0) (A.5) 

where m is a free vector that represents the velocity of translation. 

Step 2: The motion of point M is represented now in two components: ( 1 ) as translation with the velocity m and 
(2) as rotation about 00* with the angular velocity of-ot). 

Step 3: The velocity of rotation of point M about the axis 00* is determined as 

ym=^*^OM (A.6) 



NASARP-1406 35 



X 




Figure A.1 . — Representation of sliding vector. 




Figure A.2.— Substitution of sliding vector. 



36 



NASARP-1406 



Step 4: The whole velocity of point M is determined as 

V = (04^ X fi)) + (a)x OM) = fi)x (r - R) 
where r = OMand R = 04^. 

Step 5: It is easy to verify that equation (A.7) may be interpreted as 

v = 0)xp 



(A.7) 



(A.8) 



where p^r-R^ AqM represents the position vector drawn from point Aq of the axis of rotation Aq-A to point 
M, A position vect or p* m ay be drawn to M from any point on the line of action Aq-A. For instance, we may 
consider that p* = a* M (not shown in the figure) and represent v as 



Taking into account that 
we obtain 



y = a}xA*M 



smce 



A*M =A^Ao-hAoM = X*a-hAQM 
v = aixA*Af =G)x(A* a+p) = ©xp 

©x A* a = 



(A.9) 
(A.IO) 
(A.11) 
(A.12) 



because of the collinearity of vectors. 

Equations (A.7) and (A.8) enable one to determine the velocity of point M by two alternate approaches. 
Henceforth, we will use the approach that is based on the substitution of sliding vector co with an equal vector 
fl>* and the vector of translation m. 

Problem A.l. Represent analytically vector velocity v of point M by considering as given (fig. A.2) 

©=£o[Osinycosy f , a4^ = R = [-£: 0]^, OM = [xyzf 
Solution. 



v= ©x(r-R) = 



1/ J/ k/ 

cosiny cocosy 
x + E y z 



= 0) 



zsmy-ycosy 
(jc + £)cosy 
-(jc + £)siny 



(A. 13) 



NASA RP-1 406 



37 



Appendix B 
Screw Axis of Motion: Axodes 

B.l Screw Motion 

Generally, the motion of a rigid body may be represented as a screw motion — rotation about and translation 
along an axis called the axis of screw motion. 

Figure B.1.1 shows that gears 1 and 2 perform rotation about crossed axes with angular velocities £0^*^ and 
aP\ The instantaneous relative motion of gear 1 may be represented as a screw motion with parameter/? about 
the S'S axis that lies in plane O that is parallel to vectors fl)^'^ and (O^'^^, To determine the location of plane fl 
and the screw parameter/?, we use the following procedure: 

Step 1: Substitute vectors fi)^'^ and -oP-^ with equal vectors that lie in plane n and with respective vector 
moments 

mf = (O^)^ X (of, mf = [O^O^)^ x -(of^ (B.1.1) 

The subscript /indicates that vectors in equations (B.1.1) are represented in coordinate system 5^ 
Step 2: The angular velocity in relative motion 0^'^^ is represented as (fig. B.L2) 

a)f^^wf-^(-wf^) = -af^^sinYJf+(d^^ (B.1.2) 

The resulting vector moment is 

Step 3: Vector moment mi^^^ is a function of 0^0 f = Xy iy.. Our next goal is to make m^^^^ collinear to G}^*^\ 
and this requirement can be represented by 

(^)^ X (of-[0fi2)^ X dp = pcop (B.1.4) 

Drawings of figure B.1.1 show that 



0,02 =0f02 + 0,0 f (B.l. 5) 

Equations (B.1.4) and (B.l. 5) yield 

(^) xaf^-i^Jol) x(of = p(of^ (B.l. 6) 

Step 4: The determination of Ofif = XAAs based on the following transformation of equation (B.l. 6): 
38 NASA RP-1406 




Figure B.I .1 .—Screw axis of rotation. 



4'2)x 



\0^f)^ X 4'2) j_ ^^2) x[(0^)^ X 42)! = ^(^2) ^ ^U2)) ^ (B 1 7) 



Equation (B.1.7) yields 



since 



-42>[42)-(o^)J=o 



(B.1.8) 



(B.1.9) 



(B.1.10) 



The scalar products of vectors in equations (B.1.9) and (B.1.10) are equal to zero because of the 
perpendicularity of cofactor vectors. 
Vectors of equation (B.1.8) are represented as (fig. B.1.1) 

4 2) = _ft,(2) sin y jy +(©<» - a,^2) cosyjky 



= a)^"[-m2i sin y j/ +(l -/n2l cos y )ky] 



(B.1.11) 



NASA RP-1406 



39 



where m2\ ^oP^Ico^^^ 



OsOf-Xfif 



OfOi = 'E if 



a)^P = m2\(o^^^[sin y jy +cosy k^) 



(B.1.12) 
(B.1,13) 
(B.1.14) 



The orientation of 0)^^^^ and axis 5-5 of screw motion is illustrated by the drawings of figure B. 1 .2. 
Equations (B.1.8) and (B.1.11) to (B.1.14) yield 



m2i(/n2i -cosy) 



Xf — E' " 2 

•^ l-2m2iCosy + m2i 



(B.1.15) 



Step 5: The determination of the screw parameter p is based on the following transformation of 
equation (B,1.7): 



01 



,(12) 



(O^O})^ X a>fA-a>f^ '\{W2)f X ^?] = /^{^^''^f ^^'^'^^^ 



Here 



ay 



i(12) 



m),-'^r]'o 



(B.1.17) 



because of the collinearity of two cofactor vectors in the triple product. 
Equations (B.1.16) and (B.1.17) yield the following expression forp: 



p = E 



m2isiny 



l-2m2i cos 7 + ^21 



(B.1.18) 



For the case when the rotation of gear 2 is opposite to that shown in figure B . 1 . 1 , it is necessary to make m2 1 
negative in equations (B.1.15) and (B.l.lS). A negative value of X^in equation (B.1.15) indicates that plane 11 
intersects the negative axis jc^. A negative value ofp indicates that vector ;7fi)^^^^ is opposite the direction shown 
in figure B. 1.1. 




Parallel 

XOXf 

Figure B.1 .2.— Relative angular velocity. 



40 



NASA RP-1 406 



B.2 Axodes 

For the case of rotation between crossed axes with a constant gear ratio, the axodes are two hyperboloids of 
revolution. The axode of gear / (/ = 1 ,2) is formed as a family of instantaneous axes of screw motion that is 
generated in coordinate system 5- when gear / is rotated about its axis. An axode as a hyperboloid of revolution 
is shown in figure B.2. 1 . Two mating hyperboloids (fig. B.2.2) contact each other along a straight line that is 
the axis of screw motion. The relative motion of hyperboloids is rolling with sliding (about and along the axis 
of screw motion). 

In the real design of gears with crossed axes, pitch surfaces instead of axodes are applied. In cases of worm- 
gear drives and hypoid gear drives, the pitch surfaces are two cylinders and two cones, respectively (Litvin, 
1968, 1989). The point of tangency of pitch surfaces is one of the points of tangency of gear tooth surfaces. 




-Throat of 
hyperboloid 



Axis of 

screw 

motion 




Figure B.2.1.— Hyperboloid of revolution. 



Figure B.2.2.— Mating hyperboloids. 



NASA RP-1406 



41 



Appendix C 

Application of Pluecker's Coordinates and Linear 

Complex in Theory of Gearing 



C.l Introduction 

This appendix covers the basic concepts of Pluecker's coordinates, Pluecker's equation for a straight line, 
and Pluecker's Hnear complex (Pluecker, 1865). It is shown that applying the linear complex enables one to 
illustrate the vector field in the screw motion and to interpret geometrically the equation of meshing and the two- 
parameter enveloping process. 

Many scientists considered Pluecker's ideas to have been a significant contribution to the theory of line 
geometry (e.g., the work by Klein 1939; Bottema and Roth 1979; and Hunt, 1978). 

The concepts of Pluecker's coordinates and linear complex were applied to the theory of gearing by Brandner 
(1983, 1988) and Grill (1993) and Haussler et al. (1996). This section is limited to Pluecker's representation 
of a directed line, his linear complex, and the application of these to the theory of gearing. 

C.2 Pluecker's Presentation of Directed Line 

A straight line in a space is determined by a given position vector r^ of a point M^ of the line and by a directed 
vector a that is parallel to the straight line (fig. C.2. 1). The parametric representation of a directed straight line 
is 



r = r^ + M^M = ro+Xa {X^Ol XeA (C.2.1) 

The straight line is directed in accordance with the sign of the scalar factor X, 

If it is assumed that a passes through Af ^, the moment of the directed straight line with respect to origin O is 
determined as 

m^=r^xa (C.2,2) 

It is easy to verify that 

r X a = (r^ + Aa) X a = r^ X a = hIq (C.2.3) 

in^-a = (rxa)a = (C2.4) 

Equations 

mQ=rxa = roXa, in^a = (C.2.5) 

are satisfied for any current point of the straight line. These equations determine a straight line and called 
Pluecker's equations of a straight line. Therefore, six coordinates {{a^,aya^), {m^,niym^)) are called Pluecker's 
coordinates. Only four of these coordinates are independent since m^ • a = and have a common scalar factor. 
We may, for instance, consider a^ = a/lal, and m* = r x a^ instead of a and m^. 

42 NASARP-1406 




Figure C.2.1,— Representation of straight line. 



C.3 Pluecker's Linear Complex for Straight Lines on a Plane 

A plane may be determined by considering as given: (1) a point M^ through which passes the plane and (2) 
a normal N to the plane (fig. C.3.1). 
A straight line that belongs to plane n is represented as 



r = r^ + Aa (A^tO), Na = 
Equations (C.3.1) yield 

N(r-r„) = 
and the plane may be represented by the equation 

N-r = d 



(C3.1) 



(C.3.2) 



(C.3,3) 



where N, r^, and consequently, d are given. 

We consider now a line (g, m ), which belongs to the same plane if the following equation is observed 
(Merkin, 1962): 

N.(gxmJ = 4g.g) (C.3.4) 

Direct transformations of equation (C.3.4) yield 

N-(gxniJ = N.[gx(rxg)] = N-[r(g-g)-g(N-g)] = NT(g.g) = rf(g-g) (C3.5) 

We are reminded that N • g = since g must lie in plane n. 

Considering lines (a, m^) and (g, m^, we can verify that the couple of lines intersect each other or they are 
parallel if the following equation is observed: 



a m^ + m^ g = 

The proof of this statement is based on the following: 
(1) Direct transformations of equation (C,3.6) yield 

NASA RP-1 406 



(C.3.6) 



43 




Figure C.3.1 .—Representation of straight lines on a plane. 



a'(r^x^) + (r^xa)-g = (r^-r^)-(axg) = 
(2) Equation (C.3.7) is observed if 



(C.3.7) 



(a) r^ - r = 0, a x g 9t 0, which means that the Hnes have a common point. 

(b) r^ 9i r , a X g = 0, which means that the two lines are parallel. 

Equation (C.3.6) represents Pluecker's linear complex of all lines (g, mp defined by a given vector a and 
couple (a, m^). 

Line (g, m ) has six Pluecker*s coordinates (^^, 8y^ gz> ^gx^ ^gy ^g^^ ^^^ ^"^y ^^ree are independent 
because a line has only four independent Pluecker's coordinates and an additional relation between the 
coordinates is provided by equation (C.3.6). 



C.4 Application to Screw Motion 

Screw Motion 

We consider the rotation of two gears about crossed axes. The relative motion may be determined as the screw 
motion represented by vector (O^ and moment m^ = p©^ (fig, C.4.1). Here, co^ and m^ are the angular velocity 
of rotation about and translation along the instantaneous axis of screw motion; p is the screw parameter. 

Two coordinate systems 5j and Sj are rigidly connected to gears 1 and 2. Let us say that gear 1 is movable 
whereas gear 2 is held at rest. Point M of gear 1 will trace out in ^2 (in infinitesimal motion) the small arc of a 
helix . The velocity v of point M of gear 1 with respect to point M of gear 2 may be represented by vector equation 



v = o)^xr + p(0^ 



(C.4.1) 



where r is the position vector of point M. Two components of v represent the velocities in rotation about and 
translation along the screw axis. 



44 



NASA RP-1 406 



i ; Axis of screw motion 





Null plane (11) 



(a) 

Figure C,4.1.— Presentation of relative velocity in screw motion, (a) Helix and trihedron t, b, 
and c. (b) Null plane. 



Helix and Trihedron x, b, and c 

The velocity v is directed along the tangent x to the helix. Tangent T and the screw axis form the angle 
(90° -A,) (fig. C A 1) where 



(OsP P 

and p is the shortest distance of point M from the axis of screw motion. 
The helix on a cylinder with radius p is represented as 

r{e) = [pcos{e-\-q) ps\n{e-^q) pO] 



(C.4.2) 



(C4.3) 



where is the varied helix parameter; q and 6=0 determine in plane z = the location of the helix reference 
point. The vector field of velocity v may be determined as the family of tangents to helices that are traced out 
on coaxial cylinders of various radii p. Any one of the family of helices is a spatial curve, A small arc of the 
helix is traced out in Sj by point M of 5j. and it belongs to the osculating plane formed by tangent r to the helix 
and the principal normal c to the helix (fig. C.4.2), 

Trihedron x, b, and c is formed by tangent X to the helix, binormal b, and principal normal c (see definitions 
in Favard, 1957; Litvin, 1994), The unit vectors of the trihedron are represented by 



C = bXT 



(C,4.4) 



After derivations, we obtain 



T= 



-sin (0 + ^7) COS A 

cos (0 + ^) cos A 

sin A 



(C.4.5) 



NASA RP-1 406 



45 



Normal 
plain - 



Tangent 
plain 




(a) 



Null plain 




Binonnal 



Osculating plane 



(b) 



— Lr 



Principal 
nonmal 



Figure C.4.2.— Illustration of orientation of null plane, (a) Nonmal plane and tangent plane, (b) Null plane 
and osculating plane. 



b = 



sin Asin(0 + ^) 

-sin Xcos{9-¥q) 

cos A 



(C.4,6) 



c = 



-cos{6 + q) 
-sin(0 + ^) 




(C.4.7) 



Null Plane, Null Axes, and Linear Complex in Screw Motion 

We choose in coordinate system Sj a point M and determine the velocity v of point M of coordinate system 
5j with respect to the screw motion by using equation (C.4. 1). Then, we determine plane n that passes through 
M and is perpendicular to v (fig. C.4. 1 (b)). Plane U is called the null plane of point M and vice versa, and point 
M is called the null point (pole) of O. Plane U contains straight lines (g, m^) that pass through pole M. In 
accordance with the definition of plane n, we have 



vg = 



(C.4.8) 



Equations (C.4.8) and (C.4.1) yield 



m^ g + O)^ nia =0 



(C.4.9) 



where m =z7CD,m =rxg= OM x g. Equation (C.4.9) may be called Pluecker^s linear complex for screw 
motion. 

Any straight line (g, m ) through M that belongs to null plane 11 is called a null axis. Particularly, the binormal 
to the helix is also a null axis. 



46 



NASARP-1406 



C*5 Interpretation of Equation of Mesliing of One-Parameter 
Enveloping Process 

The basic form of the equation of meshing (see section 1.2) is represented by 

NrvP^=0 (/ = 1,2,/) 



(C.5.1) 



where N- is the common normal to the contacting surfaces and v/*^) is the relative velocity. The scalar product 
of vectors is invariant to the applied coordinate systems 5,, ^j, and 5^that are rigidly connected to gears 1, 2, 
and the frame, respectively. Equation (C.5. 1 ) is obtained from the conditions of tangency of the generating and 
enveloping surfaces. 

The application of the concept of Pluecker's linear complex enables one to illustrate the equation of meshing 
in terms of the null axis and the null plane and the orientation of the normal N in the null plane. We now introduce 
two approaches for determining Pluecker^s linear complex. 

Approach 1. Since normal vector N to the contacting surfaces is perpendicular to the relative velocity v , the 
normal (N, xtip^) belongs to the null plane, and an equation similar to (C.4.9) yields the linear complex 



m^ N + co^ myv =0 



(C5.2) 



where m^ = pco^ and myy = OM x N. 

The orientation of the normal (N, m^) in the null plane is shown in figure C.4.2. 

Approach 2. The concept of using screw motion for the determination of relative velocity was presented in 
Approach 1 and in section C.4. We have considered as well that coordinate system 5j rigidly connected to gear 

1 performs a screw motion with respect to coordinate system 52 rigidly connected to gear 2 while Sj and gear 

2 are held at rest. However, this concept is not the primary one to be used in kinematics and in the theory of 
gearing. The following derivations of Pluecker's linear complex are based on the concept that both gears 
perform rotations about their axes as shown in figure C.5.1. The derivation procedure follows: 

Step 1: Rotation about crossed axes is performed with angular velocities (O^^^ and (iP^ as shown in fig- 
ure C.5. 1 (a). By substituting -o^^^ with an equal vector that passes through 6y and a respective vector moment 
m^, we obtain the velocities in relative motion (fig. C.5.1(b)): 



^f 



i 
Of, 


I 




/ 


/ 


i 


02, 




E 


/ 


V / 




(a) 

Figure C.5.1 .—Representation of normal to contacting surfaces as the null axis, (a) Applied coordi- 
nate system and vectors «(^) and «(2) of angular velocities, (b) Relative angular velocity wC^)^ 
vector m^^ and surface nonmal N. 



NASA RP-1406 



47 



CO 



= a>(*^ - co(2\ m^ = E X (-co^^))^ (^ , ^(12) J (^.5.3) 



where E = Of02 . 



Step 2. It is known from differential geometry and the theory of gearing that the generating surface and the 
envelope are in line contact at every instant. Figure C.5. 1 (b) shows that p oint M is a point of the line of contact, 
N is the common normal to the contacting surfaces at M, and r = OfM is the position vector of M, 

The normal as a straight line may be represented by the direction vector N and the respective vector moment 
niyy. Thus, we consider as given 

N, m;v=rxN (C5.4) 

If (N, ndy^) belongs to Pluecker's linear complex, the following equation must be observed: 

m^ • N + m;v • CO = (C.5.5) 

Equations (C.53) to (C.5.5) yield 

-(e X co^^^) • N + (r X N) • (O = (C.5.6) 

After transformations of equation (C.5.6), we obtain 

N • [(a)('2) X r) - (e X co^^))] = N ■ v^^) = o (C.5.7) 

Equation (C.5.7) is observed since it is the equation of meshing of the contacting surfaces. This means that 
N is the null axis. 

Using Pluecker*s terms, we can determine the null plane as the one that passes through contact point M, 
contains the normal N, and is perpendicular to the moment. 

m* = -(Ex(D(2)) + [-rx((o(')-a)<2))] = ((o<')-a)(2))xr-(Ex(D(2)) (C.5.8) 

It is evident that m* = v^*^\ which means that the null plane is perpendicular to the relative velocity v^*^\ 
The drawings of figure C.4.2(a) illustrate the concept of planar L and spatial L^ curves located on a tooth 
surface. Both curves are in tangency at point M and their common unit tangent is designated T. These drawings 
also show the tangent plane to surface X at point M. 

The drawings of figure C.4.2(b) represent in an enlarged scale curves L and L^. For the case of the meshing 
of the envelope and the generating surface, spatial curve L^ is the trajectory that is traced out in relative motion 
by point M of the moving surface on the surface held at rest. The relative motion is a screw motion (see section 
C.4), and the relative velocity is directed along the tangent to the helix at point M that coincides with the unit 
tangent T shown in figure C.4.2(b). The null plane at any point M of the tangency of the envelope and the 
generating surface can be determined as follows. Consider as known at point M the common normal N to the 
surfaces and the vector of relative velocity or its unit tangent T. Then, the null plane at point M is determined 
to be the plane that passes through normal N and is perpendicular to T, the unit vector of relative velocity 
(fig. C.4.2(b)). We are reminded that the null plane also contains the binormal and the principal normal to the 
trajectory L^ at point M. 



48 NASA RP-1 406 



C.6 Interpretation of Equations of Meshing of Two-Parameter 
Enveloping Process 

Introduction 

In a two-parameter enveloping process, the generated surface is determined to be the envelope of the two- 
parameter family of surfaces (Litvin, Krylov, and Erikhov, 1975; Litvin and Seol, 1996). 

The generation of an involute helical gear with a grinding worm is used to illustrate the two-parameter 
enveloping process. The method of grinding the spur and the helical gears with a cylindrical worm and the 
grinding equipment were developed by the Reishauer Corporation, The meshing of the worm with the gear 
being ground may be considered as the meshing of two involute helicoids with crossed axes (Seol and Litvin, 
1996a). 

Figure C,6.1 shows the grinding worm and the.gear to be ground and the shortest distance between the axes 
as the extended one (for the simplification of the drawings). The machining center distance is installed as 



^wg - ^pw + ^pg 



(C.6.1) 



where r^^ and r are the radii of the pitch cylinders of the worm and the gear. With such a center distance, points 
M, and Afj of the pitch cylinders will coincide with each other and the pitch cylinders will be in tangency. 



Helical gear 




Grinding worm 



Rgure C.6.1 .—Representation of grinding of helical 
gear by womri. 



NASA RP-1 406 



49 



Figure C.6.2 shows the coordinate systems applied to describe the generation process. The movable 
coordinate systems S^ and S are rigidly connected to the worm and the gear, respectively. Coordinate systems 
S^, 5a and 5^ are fixed. 

For the case when the worm and the gear are both right hand, the crossing angle y between the axes of the 
worm and the gear is represented by 



Ywg ~ ^pw "*" ^pg 



(C.6.2) 



where A • (/ = w,g) is the lead angle on the pitch cylinder. 
The process of gear generation requires two independent sets of motion: 

(1) The first set of motions is executed as the rotation of the gear and the worm about axes Zg and z^, 
respectively, with angular velocities co^^ and co^^^ related by 




Figure C.6.2. — Coordinate systems for grinding of helical gear by worm. 



50 



NASA RP-1406 



co^^^ 



CD 



is) 



= ^g (C.6,3) 



It is assumed that a one-thread grinding worm will be used; N is the number of the gear teeth. 

(2) Th e secon d set of motions is executed as the translation of the worm with the velocity d\/^Vdt in the 
direction O (9^, which is parallel to the gear axis, and the additional rotation of the gear with the angular 
velocity Q^>. Here, 

'^-^fi'^' (C.6.4, 

where /?^ is the gear screw parameter (p^>0 for a right-hand gear). The second set of motions d\|/^Vdt and Q^^ 
is required to provide the feed motion during the grinding process. 

Derivation of Gear Tooth Surface 

The gear tooth surface Z^ is an envelope of the two-parameter family of worm thread surfaces and is 
determined by the following equations (Litvin, Krylov, and Erikhov, 1975): 

r^ (w, e, 0, y/) = Mg^ (0, if/)r^ («, 0) (C.6.5) 

N(->.v|-^'0>=O (C6.6) 

^M.y(-8^^)=0 (C.6.7) 

where iu,9) are the surface parameters of the worm thread surface I^; and v^are the generalized parameters 
of motion of the two sets of independent motions mentioned above; r (w,a,0,vO is the vector function that 
determines in S the two-parameter family of surfaces E^ generated in f ; v(^^'^> is the relative velocity that 
is determined when the variable is the parameter of motion 0and y/ is held at rest; v^**'«*V^ is the relative velocity 
in the case when the variable is the parameter of motion v^and <p is held at rest; (C,6.6) and (C.6.7) are the 
equations of meshing and the subscript / = g,n,mj,w indicates that the respective scalar product of vectors is 
invariant to the applied coordinate system. 
Equations (C.6.5) to (C.6.7) represent the sought-for tooth surface by four related parameters. 

Interpretation of Equations of Meshing by Application of Pluecker's Linear Complexes 

The transformation of the equation of meshing (C.6.6) is based on the following: 

(1) The gear and the worm perform rotational motions about the crossed z^- and z -axes with angular 
velocities ©(**') and a)^\ respectively (fig. C.6.2). Vectors of the scalar product are represented in the 5. 
coordinate system. ^ 

(2) The relative velcoity y^**'^'^) is represented as 

v^"^^^> = (a></> -^f)xrf~Ojd'gX (d(^> (C.6,8) 

Then we obtain the following equation of meshing (C.6,6): 

[((ojr^ - G}f) XTf- Ojo'g X a)^^> j . N^^> = (C.6.9) 

where r^ is the position vector of the point of tangency of surfaces I^ and I . 
We interpret the equation of meshing (C,6.9) by Pluecker's linear complex using the following procedure: 

Step 1: The relative motion during generation is represented by vector a)and moment m^ that are represented 
as 

NASARP-1406 51 



Step 2: The common normal N to the contacting surfaces and the moment m^ of N represented as 

myv = 0^xN = ryxN (C6.ll) 

are related to CO and m^ by the equation of the linear complex 

G)-myv+in^-N = (C6.12) 

Equation (C.6. 1 2) yields the equation of meshing (C.6.9). The null plane passes through the point of tangency 
Af, contains the normal N, and is perpendicular to v^^^'*^\ The normal N is the null axis of the linear complex 
(see section C.3). 

Let us now consider the interpretation of the equation of meshing (C.6.7) by application of Pluecker's linear 
complex. The second set of the paremeters of motion is represented by the couple of vectors (A m^), which 
can be obtained from equations that relate the motion of the worm with respect to the gear. Here, 

Q = Q^"^^ - Q^^^ = -Q^^^ (C.6. 1 3) 

since ^2^^) = 0, 

ma=^^ (C.6.14) 

at 

The other couple of vectors is represented by 

N, mj,! 

where m^^ is represented by equation (C.6.1 1). 
Both couples of vectors are related by the equation of the linear complex: 

i2-my^+m^-N = (C.6.15) 

that yields 

-d^^-[rfXN) + ^ N = (C.6.16) 

Equation (C.6.16), after simple transformations, may be represented as 






N = (C.6.17) 



Equations (C.6.17) yields 

yi^^s.¥),^M=Q (C6.18) 

The null plane for the second set of parameters of motion passes through the same point M and is 
perpendicular to y^^s^^ but not to \^^sS, The conmion surface normal N is the null axis of the second linear 
complex. 



52 NASA RP-1 406 



Chapter 2 

Development of Geometry and Technology 

2.1 Introduction 

Errors in gear alignment and manufacture may shift the bearing contact, turn it into edge contact, and cause 
transmission errors that, as we are reminded, are the main source of vibration. The purpose of this chapter is 
to present the latest developments in gear geometry and technology directed at improving bearing contact and 
reducing transmission errors. 

The main errors of alignment and manufacture are as follows: the error of the shaft angle, the shortest center 
distance, the leads in the case of helical gears, the errors in machine-tool settings (errors of orientation and 
location of the tool with respect to the gear being generated), and the errors of the circular pitches. In addition, 
we have to take into account the deflection of the teeth and shafts under load. To avoid or at least to reduce such 
defects, it becomes necessary to substitute the line contact of the gear tooth surfaces by the point contact and 
then, in addition, to modify the gear tooth surfaces. The modification of gear geometry is based on the proper 
deviation of the gear tooth surfaces from the theoretical ones. The surface deviation can be provided ( 1 ) in the 
longitudinal direction with the contact path in the profile direction (the direction across the tooth surface) and 
(2) in the profile direction with the longitudinal direction of the contact path. In some cases, both types of 
deviation must be provided simultaneously, but one of them must be the dominant. 

The desired modification of gear geometry becomes possible by applying inventive methods of gear 
technology such as (1) the mismatch of tool surfaces for the generation of spiral bevel gears and hypoid gears, 
(2) the varied plunge of the tool for the generation of spur and helical gears, (3) the application of an oversized 
hob for the generation of worm gears. Some of these examples are considered in the following sections. We 
emphasize that in all of such cases of gear manufacture, it is important to provide a predesigned parabolic 
function of transmission errors and to reduce their magnitude (see section 1.14), which will improve the 
conditions of the transfer of meshing while one pair of teeth is changed for the neighboring one. 

This chapter summarizes the developments achieved at the Gear Research Laboratory of the University of 
Illinois at Chicago. Details are given in Litvin and Kin (1992); Litvin and Hsiao (1993); Litvin and Lu (1995); 
Litvinetal.(1995, 1996a, 1996b);Litvin, Chen, and Chen(1995);Litvin and Feng (1996, 1997); Litvin, Wang, 
and Handschuh ( 1 996); Litvin and Seol ( 1 996); Seol and Litvin ( 1 996a, 1 996b); Zhang, Litvin, and Handschuh 
(1995); and Litvin and Kim (1997). 

2.2 Modification of Geometry of Involute Spur Gears 

Localization of Contact 

Spur gears are very sensitive to the misalignment of their axes, which causes an edge contact. The sensitivity 
of the gear drive to such a misalignment can be reduced by localizing the bearing contact. The localization of 
the contact as proposed in Litvin et al. (1996b) can be achieved by plunging the grinding disk in the generation 
of the pinion by form-grinding (fig. 2.2.1). The mating gear is generated as a conventional involute gear. The 
plunging means that during the pinion generation, the shortest distance between the axes of the grinding disk 
and the pinion will satisfy the equation (fig. 2.2.1) 

NASARP-1406 53 



Grinding 
wheel 




Gear 



Pinion 



Figure 2.2.1 .—Form-grinding of pinion 
by a plunging grinding wheel to satisify 
E = Eq~ Sfjlfj, where E and Eq are the 
cun^ent and nominal values of the shortest 
distance, a^ is the parabola coefficient of 
function E(IJ), and /^ is the displacement 
of the disk in the direction of the pinion 
axis. 






_ Pinion 




-2 



-1 



Tooth width 



Figure 2.2.2.— Path of contact and bearing contact 
of aligned gear drive with modified geometry. 



E-E^-aJ^ 



(2.2.1) 



where E and E^ are the current and nominal values of the shortest distance, a^ is the parabola coefficient of 
function E(ljX and /^ is the displacement of the disk in the direction of the pinion axis. The localized bearing 
contact is shown in figure 2.2.2. ^ 

The same results can be obtained by varying the shortest distance between the hob (grinding worm) and the 
generated spur or helical pinion. This variation is based on the application of an equation that is similar to 
equation (2.2.1). 

Transformation and Reduction of Transmission Errors 

The profile of the pinion cross section for lj=0 coincides with the axial profile of the disk. If the axial profile 
of the disk is designed as the involute profile of the conventional pinion, the gear drive is still sensitive to 
misalignment ^aand the error of tooth distance, which will cause an almost linear discontinuous function of 
the transmission errors with the period of the meshing cycle 0i = iTt/Ni (fig. 2.2.2), Applying a predesigned 
parabolic function enables the absorbtion of the transmission error linear functions shown in figure 2.2.3 (see 
section 1.14). 

The predesign of the parabolic function of the transmission errors is based on the following alternative 
approaches: (1) changing the curvature of the pinion or the gear and (2) executing a nonlinear function that 
relates the rotational motion of the gear (or the pinion) and the translation of the imaginary rack-cutter used to 
generate the gear (or the pinion). 

Changing the curvature of the pinion can be achieved by substituting the tooth involute profile that 
corresponds to the theoretical base circle of radius r^i by an involute profile of radius r^i (fig. 2.2.4). 



54 



NASARP-1406 




10 20 

Pinion rotation, (t>^, deg 



30 



Figure 2.2,3. — ^Transmission errors caused by errors of 
pressure angle Aa = 3 arc min. 




Figure 2.2.4,— Theoretical and modified profiles of spur pinion. 



NASA RP-1406 



55 



23 Modification of Involute Helical Gears 

Computerized Investigation of Misaligned Conventional Involute Helical Gears 

Aligned involute helical gears are in line contact at every instant, as shown in figure 2.3.1. Misalignment 
caused by changes in the shaft angle, the lead, and the normal profile angle of one of the mating gears causes 
edge contact instead of surface-to-surface tangency . Edge contact means tangency of the edge of one gear with 
the tooth surface of the mating gear (see section 1.11). The edge and the surface are in mesh at every instant at 
a point instead of on a line. An example of an edge contact caused by the change of the shaft angle Ay or the 
change of the pinion lead angle AXp^ is shown in figure 2.3.2. The edge contact caused by Ay or AXpi is also 
accompanied by transmission errors, as shown in figure 2.3.3. However, the change in the normal profile angle 
does not cause transmission errors, only an edge contact. 

In the case of a change in the center distance £", the gear tooth surfaces are still in a line contact similar to those 
shown in figure 2.3.1, However, an error in AE causes a change in the backlash and in the pressure angle of the 
gear drive. 

There is a mistaken impression that a change in the lead is sufficient to shift the bearing contact from the edge 
to the central position and avoid transmission errors. Our investigation shows that a combination of ^y- and 
A?ipi-ervoTS will enable one to avoid transmission errors and obtain the favorable contact path shown in figure 
2.3.4 ifand only if 

^7 = |^A^i| (2.3.1) 

The signs of ^yand AX^i depend on the hand of the helix. 

Equation (2.3.1) must be observed with great precision because even a small difference between ^yand 
\AXpi\ (or \AXpj) will cause an edge contact. Therefore, the change of the lead, if it is not accompanied by 
applying a predesigned parabolic function of transmission errors (see below), is not a convenient way to avoid 
an edge contact. In conclusion, we must emphasize that the computerized investigation of misaligned helical 
gears is a complex mathematical problem because the Jacobian of the system of equations that relate the surface 
parameters and parameters of motion is close to zero (see section 1,11). What is required for the computerized 
investigation of a misaligned gear drive is a determination of the proper initial guess. 



Contact lines - 




cylinder 
helix 



Figure 2.3.1 .—Contact lines on tooth surfaces of 
helical gear. 

56 NASARP-1406 





Figure 2.3.2.— Edge contact caused by Ay or AXpi 
3 arc min. (a) Pinion tooth surface, (b) Gear tooth 
surface. 



18 



17 



16 



i 


15 


^ 




g 


14 


o 




c 




.2 


13 


S 





E 12 



11 




J \ I 



10 

-15 -10 -5 5 10 15 20 25 
Pinion rotation, <>-|, deg 

Figure 2.3.3. — Function of transmission errors caused by 
Ay or AXpi = 3 arc mIn. 




(b) 

Figure 2.3.4.— Path of contact caused by A7 = 3 arc min 
and AXp^ = -3 arc min. (a) Pinion tooth surface, (b) Gear 
tooth surface. 



NASA RP-1406 



57 




(a) 





Figure 2.3.5.— Normal section of rack-cutters, (a) Rack-cutters for gear and pinion generation, (b) Pinion 
rack-cutter, (c) Gear rack-cutter. 




Figure 2.3.6.— Path of contact caused by A7 = 3 arc min. 
(a) Modified pinion tooth surface, (b) Gear tooth surface. 



58 



NASA RP-1406 



20 



o 


15 


o 




in 




<) 






10 


CM 




< 


5 


\Z 




§ 





o 




c 




o 


-5 


0) 




» 






-10 


c 




^ 




1- 


-16 



-20 




J \ \ L 



J \ L 



-25 -20-15-10-5 5 10 15 20 
Pinion rotation, <|>t, deg 

Figure 2.3.7.— Function of transmission errors for modified 
involute helical gear drive when A7 = 3 arc min. 



Profile Modification 



The sensitivity of helical gears to misalignment has led designers and manufacturers to crown tooth surfaces. 
The most common method of crowning is profile modification; that is, the profile of the cross section of one 
of the mating gears (usually of the pinion) is deviated from the conventional involute profile. This can be 
achieved as proposed in Litvin et al. (1995) by the application of two imaginary rack-cutters shown in 
figure2.3.5. The rack-cutters are rigidly connected and generate the pinion and the gear separately.Figure2.3.5(a) 
shows both rack-cutter profiles. The normal section of the pinion rack-cutter that generates the pinion space is 
shown in figure 2.3.5(b), and the normal section of the gear rack-cutter that generates the gear tooth is shown 
in figure 2.3.5(c). The deviation of the pinion rack-cutter profile from the gear rack-cutter profile is represented 
by a parabolic function with the parabola coefficient a^. The tooth surfaces in the case of profile modification 
are in point contact, and the path of contact is a helix, as shown in figure 2.3.6. 

It can be easily verified that the profile modification enables one to localize the bearing contact and to avoid 
an edge contact that might be caused by gear misalignment. However, the discussed modification does not allow 
the elimination of transmission errors caused by misalignment, as shown in figure 2.3.7. Therefore, to reduce 
the level of noise and vibration, it is necessary to provide a predesigned parabolic function of transmission errors 
in addition to profile modifications. 



2.4 Development of Face-Gear Drives 

Introduction 

Face-gear drives have found application in the transformation of rotation between intersected and crossed 
axes. The Fellows Corporation invented the method of manufacturing face gears by a shaper which is based on 
simulating the meshing of the generating shaper with the face gear by cutting. 

Face-gear drives was the subject of intensive research presented in Davidov (1950) and Litvin (1968, 1994). 
New developments in this area were supported by NASA Lewis Research Center and McDonnell Douglas 
Helicopter Systems. An important application of face-gear drives in helicopter transmissions is based on the 
concept of torque split (fig. 2.4. 1 ). There are other examples of the successful application of face-gear drives 
in transmissions. 



NASA RP-1 406 



59 



Rotor shaft output 



Sun gear 



Combining 
gear— ^.^ 



notartm 

output 




\ / 
"^ ' \ / 

^ Face-gears ^^gj^^ -^^^^^ __j^ 
Figure 2.4.1 . — Application of face-gear drives in helicopter transmissions. 



Pitch Surfaces 

The pitch surfaces for bevel gears are two cones (fig. 2.4.2(a)) with pitch angles /i and 7^. The line of tangency 
01 of the pitch cones is the instantaneous axis of rotation. The cones roll over each other in relative motion that 
can be represented as rotation about OL The pitch surfaces in a face-gear drive are the pinion pitch cylinder of 
radius r^x ^"d the face-gear pitch cone with the pitch angle 7(fig. 2.4.2(b)). The tooth element proportions in 
bevel gears are related by the application of the pitch line 01 (fig. 2.4.2(a)) as the middle line of the teeth. The 
tooth height in face-gear drives is constant, and the middle line of the teeth is the line of tangency 0*M of the 
pinion pitch cylinder of radius r^i and the gear pitch cone with apex angle y. It will be shown in the next section 
that since 0*M does not coincide with the instantaneous axis Oh the face-gear teeth become sensitive to 
undercutting and pointing. 

Generation of Face Gears 

The process of generation by a shaper is shown in figure 2.4.3. Grinding and bobbing by a worm is a recently 
developed process for face-gear generation. Methods for determining the surface of the worm (fig. 2.4.4) and 
its dressing were developed by researchers at the University of Illinois at Chicago and at McDonnell Douglas 
Helicopter Systems. 

Avoidance of Undercutting and Pointing 

The tooth of a face gear in three-dimensional space is shown in figure 2.4.5. The fillet surface is generated 
by the top edge of the shaper tooth. Line L* is the line of tangency of the working part of the tooth surface and 
the fillet. Undercutting may occur in plane A and pointing in plane B. The tooth surface is covered by lines of 
tangency L of the face gear with the generating shaper. 

The tooth of a face gear may be undercut and pointed in the process of generation by a shaper. These defects 
can be avoided by properly designing the tooth length of the face gear. Dimensions Li and L^ determine the zone 
that is free of undercutting and pointing. The equations for computing Lj and Li are represented in Litvin ( 1 994). 
The length of teeth (Li - L2) with respect to the diametral pitch Pj may be represented by the unitless coefficient 



60 



NASA RP-1 406 




^-a 



Y, = 180«-7 




(a) (b) 

Figure 2.4.2.— Pitch surfaces of bevel gears and gears of face drive, (a) Bevel gears, (b) Face gears. 



2 (face gear) 



s (shaper) 





^^^mm^ 






02.0s.0„, 




^k' 


^m^^s 


1 


•kmi 




«{2)/ 


/ 


gL 




/ 


^^y^m 


-"•— 




l\ 









^2 Ym 
Rgure 2.4.3. — Generation of face gear by shaper. 




Rgure 2.4.4. — Grinding womri for face gears. 



NASA RP-1406 



61 



Fillet 
surface 



L*- 




Figure 2.4.5. — Tooth of face gear in three-dimensional 
space. 



c = {L2-Li)Pd 
whose value depends on the gear ratio mi2 = A^2/A^i of the face gear. It is recommend that mi2 > 4 be used to obtain 

oia 

Localization of Bearing Contact 

When the number of shaper teeth equals the number of pinion teeth, misalignment may cause the separation 
of the face-gear teeth and the pinion and then the loss of their edge contact. To avoid this defect, it becomes 
necessary to localize the bearing contact between the pinion and the face gear. This localization can be achieved 
by applying a shaper with the tooth number N^ > N^ so that N^-Np= 1-3, where Np is the pinion tooth number. 
Another approach is based on varying the tool plunging (grinding disk or cutter) in the pinion generation process 
(see section 2.2). In such an approach, A^^ = Np. 



62 



NASA RP-1 406 




^— Major axis of 
contact ellipse 



Figure 2.4.6.— Localized bearing contact; AA/ = 3; 
AE=0.1 mm. 



Results of Tooth Contact Analysis (TCA) 

A computer program has enabled researchers to investigate the influence of the following alignment errors: 
AE, the shortest distance between axes that are crossed but are not intersected; Aq, the axial displacement of the 
face gear; Ay, the change in the shaft angle formed by intersected axes. It was discovered that such alignment 
errors do not cause transmission errors, which is the great advantage of using a face-gear drive with an involute 
pinion. However, such errors cause a shift in the bearing contact, as shown in figure 2.4.6. If the pinion is 
generated by a plunged tool (see sec. 2,2). there is a good possibility of compensating for this shift due to the 
axial displacement of the pinion. 



2.5 Development of Geometry of Face-Milled Spiral Bevel Gears 

Introduction 

An important contribution of The Gleason Works engineers is the development of spiral bevel gear drives 
and hypoid gear drives with localized bearing contact and parabolic- type transmission errors (Stadtfeld, 1993, 
1995). The research conducted at the Gear Research Laboratory of the University of Illinois at Chicago (Litvin 
et aL, 1996b; Litvin, Wang, Handschuh, 1996; Zhang, Litvin, and Handschuh, 1995) was directed at the 
modification and improvement of the existing geometry of spiral bevel gears. The developed projects covered 
two types of face-milled spiral bevel gears: uniform teeth and tapered teeth. 

Modification of Geometry of Spiral Bevel Gears With Uniform Teeth 

Theoretically, ideal spiral bevel gears with zero transmission errors can be generated if the following 
conditions are observed: 

( 1 ) Two imaginary generating surfaces are rigidly connected to each other and separately generate the pinion 
and gear tooth surfaces. The generating surfaces are mismatched but they are in tangency along a line. The 
generating surfaces produce spiral bevel gears with a constant tooth height. 

(2) Two types of bearing contact are provided by this method of generation: 

(a) The mismatched generating surfaces X,, and L^ are two cones in tangency along a common generatrix 
(fig. 2,5.1), The difference in mean radii Rp and R^ determines the mismatch of the generating surfaces. The 
bearing contact is directed across the tooth surfaces. 

(b) The mismatched generating surfaces I^ and I^i are a cone and a surface of revolution in tangency along 
a circle of radius R^ = R^ (fig. 2.5.2). The axial section of the surface of revolution is a circle of radius R^, The 
bearing contact of the generated spiral bevel gears is directed along the tooth surfaces, 

NASA RP-1 406 63 



Common 
I generatrix 





Figure 2.5.1 .—Application of two generating cones. 



Figure 2.5.2. — Application of generating cone and 
generating surface of revolution. 



These generation methods provide conjugated tooth surfaces that are in point tangency at every instant. 
However, in reaHty such methods cannot be applied because the generated spiral bevel gears are sensitive to 
misalignment that will cause a shift in bearing contact and the transmission errors of the type shown in 
figure 1.14.1(b). Such transmission errors are the source of vibrations. 

Low-Noise Spiral Bevel Gears With Uniform Tooth Height 

Litvin, Wang, and Handschuh (1996) show that the defects in gearing just discussed can be avoided by the 
application of the generation method which requires that a couple of the generating surfaces be in point contact 
and that the surfaces be properly mismatched (fig. 2.5.3). The gear generating surface is a cone, and the pinion 
generating surface is a surface of revolution. The gear cutting blades are straight-line blades (fig. 2.5.4); the 
pinion cutting blades are circular arcs (fig. 2.5.5), The method of local synthesis developed in Litvin (1994) and 
applied for spiral bevel gear generation enables us to determine those design parameters of the generating 
surfaces and machine-tool settings that provide a predesigned parabolic function of transmission errors and a 
stable bearing contact. Figure 2.5.6 shows transmission errors caused by Ay= 3 arc min when the generating 
surfaces are in line contact but are not in point contact and the surfaces are not mismatched to provide a 
predesigned parabolic function of transmission errors. Figure 2.5.7 shows the resulting transmission error 
function as the interaction of the predesigned parabolic function and the transmission error function caused by 
Ay . The predesigned parabolic function was obtained by observing the following conditions: ( 1 ) the generating 
surfaces are in point contact (but are not in line contact); (2) the surfaces are properly mismatched to provide 
a predesigned parabolic function of transmission errors. Figure 2.5.8 shows the bearing contact for a misaligned 
gear drive. 

Low-Noise Spiral Bevel Gears With Tapered Teeth 

The geometry of low-noise spiral bevel gears with tapered teeth is presented in Zhang, Litvin, and Handschuh 
(1995). The authors of the project proposed an approach that was based on the application of two generating 
cones being in point contact. By applying the proposed method of local synthesis, the authors determined design 
parameters of the generating surfaces and machine-tool settings that enabled them to obtain a localized bearing 
contact and a predesigned parabolic function of transmission errors of a low level, 8 to 10 arc sec. 

Prototypes of these gears were manufactured by the Bell Helicopter Company and were then tested at The 
Gleason Works and at the NASA Lewis Research Center. The test results proved that the noise level was reduced 
by 1 8 dB from the total level of 90 dB in comparison with the existing design (fig. 2.5.9). 



64 



NASA RP-1406 




Figure 2.5.3.— Mismatched generating surfaces in 
point contact. 



Generating 
cones — is^ 

I \ 



Cutter 
/ blade 




Figure 2.5.4. — Gear generating cones. 





Figure 2.5.5.— Convex and concave sides of generating blades and pinion generating surfaces of 
revolution. 




-50-40-30-20-10 10 20 30 40 50 
Pinion rotation, <|>t, deg 

Figure 2.5.6.— Transmission en'ors for misaligned 
gear drive; A7 = 3 arc min. 




-60-50-40-30-20-10 10 20 30 40 50 
Pinion rotation, <(>^, deg 

Figure 2.5.7. — Transmission errors for misaligned 
gear drive with mismatched gear tooth surfaces; 
A7 = 3 arc min. 



NASA RP-1 406 



65 



5 r- 




Figure 2.5.8.— Longitudinal bearing contact 
for misaligned gear drive; A7 = 3 arc min. 



£4 



D) 



C 

o 



0) 

o 

T3 

i 

I 

E 



▼ Spiral-bevel mesh 
frequencies 

Former design 

New design 



k 



jt;:: : 




yi.^ii^w^^^ 



2000 4000 6000 8000 10 000 
Frequency, Hz 

Figure 2.5.9. — Results of noise reduction. 



2.6 Modification of Geometry of Worm-Gear Drives With Cylindrical 
Worms 

Introduction 

Worm-gear drives with cylindrical worms are applied for the transformation of rotation between crossed axes 
when a large gear ratio is required. The worm is similar to a helical gear whose number of teeth is equal to the 
number of threads on the worm. Figure 2.6.1 shows the cross section of a three-thread worm. The existing 
geometry of gear drives provides at every instant a line contact between the ideal surfaces of the worm and the 
worm gear. This contact can be achieved by simulating the meshing of the hob and the worm gear being 
generated. However, in reality a line contact cannot be provided because of gear drive misalignment and 
manufacturing errors. Errors of alignment and manufacture cause the surfaces to be in point contact instead of 
line contact accompanied by a shift of the bearing contact to the edge (fig. 2.6.2) andtransmission errors of the 
shape shown in figure 1.14.1(b). In the case of multithread worm drives, such transmission errors may cause 
impermissible vibrations. A suitable bearing contact can be obtained by running the worm-gear drive under a 
load while the hardened worm removes particles of materials from the much softer surface of the worm gear. 
However, such a process is time consuming, may not substantially reduce the transmission errors, and does not 
change their shape. These are the reasons for modifiying the existing geometry of worm-gear drives (Seol and 
Litvin, 1996a, 1996b). 

Brief Description of Existing Geometry of Worm-Gear Drives 

The existing types of worm-gear drive geometry may be divided into two groups: (1) worms that are 
generated by blades and (2) worms that are ground (Litvin, 1994). Figures 2.6.3 and 2,6,4 show the installment 
of straight-line blades for the generation of ZA- and ZN-worms, respectively. The blades are installed in the 
axial section (for ZA-worm generation) and in the normal section (for ZN-worm generation). 

The ground worms may be differentiated thus: Zl-worms (involute), which may be ground by a plane and 
are manufactured by the David Brown Company; ZF-worms, which are discussed separately in the next 
paragraph; and ZK- worms, which are cut or ground by a cone (fig. 2.6.5). The installment of the generating tool 
with respect to the worm is shown in figure 2.6.6. During the process of generation, the worm performs a screw 
motion about its axis with parameter/?. The installment angle y^ is usually chosen as y^ - ^, where Xp is the worm 
lead angle. 



66 



NASARP-1406 




Figure 2.6.1 .—Cross section of three-thread worm. 




Figure 2.6.2. — Shift of bearing contact in misaligned worm-gear drive. 




Figure 2.6.3.— Generation of ZA-worms. 




Figure 2.6.4. — Generation of ZN-worms. 
(a) Thread generation, (b) Space generation. 



NASA RP-1 406 



67 





Figure 2.6.5. — Tool for generation of ZK-worms. 



The generation of the ZF- worms (Flender worms) is based on the application of a grinding disk whose axial 
section is represented in figure 2.6.7(a). The axial profile of the tool is a circular arc. The worm in the process 
of generation performs a screw motion about its axis. The line of tangency between the surfaces of the grinding 
disk and the worm surface is a spatial curve. However, this line of tangency might be a planar curve that 
coincides with the axial profile a-a (fig. 2.6.7(b)) if special machine-tool settings are provided (proposed in 
Litvin, 1968). 

Modification of Geometry of Worm-Gear Drives 

Worm-gear drives with the existing geometry are very sensitive to misahgnment. Errors in misalignment 
cause a shift in the bearing contact, as shown in figure 2.6.2. This defect can be avoided by applying an oversized 
hob (Colbourne, 1993; Kovtushenko, Lagutin, and Yatsin, 1994; Litvin et al., 1996b; Seol and Litvin, 1996a, 
1996b).The principles of this application follow: 

(1) The oversized hob and the worm are considered as two helical gears in internal tangency (fig. 2.6.8). 
Conjugated surfaces of the hob and the worm can be provided because they are simultaneously in mesh with 
a common rack. 

(2) The application of an oversized hob enables one to localize the bearing contact since the surfaces of the 
worm and the worm gear generated by an oversized hob are at every instant in point contact, not in line contact. 
The bearing contact is localized and is in the middle area of the worm-gear tooth surface (fig. 2.6.9). 

The localization of bearing contact in Flender worm-gear drives does not guarantee a reduction in magnitude 
and a favorable shape of transmission errors (fig, 2.6, 10(a)). It was shown in Seol and Litvin (1996a) that a 
parabolic function error for Flender worm-gear drives (fig. 2.6.10(b)) can be provided by varying the process 
for generating the distance between the axes of the hob and the worm gear or by modifying the shape of the 
grinding disk. A coipputerized investigation of a misaligned worm-gear drive with surfaces in line 
contact is a complex problem, as explained in section 1,11. 

A computerized investigation of a worm-gear drive generated by an oversized hob requires two stages of 
computation: ( 1 ) a determination of the worm-gear tooth surface generated by the oversized hob, a case in which 
the hob and worm-gear surfaces are considered to be in line contact; (2) a simulation of the meshing and contact 
of the worm-gear tooth surface and the worm thread that are in point contact. Such computer programs were 
developed by Seol and Litvin (1996a, 1996b). 



68 



NASA RP-1 406 




•e ' 









\ 






V 






\ 






>^^«^ 


J\ 




^frc 




^^ " ^^^^ 




^" y 



Figure 2.6.6.— Tool installment and generation of ZK-womis. 





Figure 2.6.7.— Grinding of ZF-(Flender) wonns. (a) Installment of grinding disk, (b) Disk axial profile. 



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69 



Worm 




Figure 2.6.8.— Pitch cylinders of oversized hob and worm. 




Figure 2.6.9.— Path of contact and bearing contact of 
misaligned Zl-worm-gear drive; AE = 0.1 mm; A7 = 
1 .0 arc min. 





§0 

« s 

c «> -10 
.2 P 

M (S 
<0 . 

1 J? -20 



-30 



(a) 



2WWi 
Pinion rotation, <t>i, deg 




(b) 



k-2«/A/i»l 
Pinion rotation, <t>i, deg 



Figure 2.6.1 0.— Transmission en-ors of misaligned, modified Render worm-gear drive, (a) No plunging, 
(b) Plunging (a = 0.03; A£ = 0.1 mm; A7 = 3.0 arc min). 



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Figure 2.7.1 .—Face worm-gear drive with intersected axes. 



2.7 Face Worm-Gear Drives 

The face-gear drives discussed in this section were formed by a cylindrical or conical worm and a face gear. 
These types of gear drives were invented by O.E. Saari and are described in Saari (1954, 1960). They were the 
subject of research conducted by the inventor and other researchers whose results are presented in Goldfarb and 
Spiridonov (1996), Kovtushenko, Lagutin, and Yatsin (1994), and in many other papers. This section presents 
the results of research conducted at the Gear Research Laboratory of the University of Illinois at Chicago by 
F.L. Litvin, A, Egelja, and M. De Donno. The goals of the research projects were ( 1 ) to provide a computerized 
design that enables one to avoid the undercutting and pointing of face worm gears, (2) to effect the localization 
of bearing contact, and (3) to accomplish a reduction in magnitude and a transformation of the shape of the 
transmission error function into a favorable one (see section 1.14). The cause of the the transmission errors was 
considered to be misalignment. Special attention was given to the simulation of meshing and the contact of 
misaligned gear drives by developed TCA (Tooth Contact Analysis) computer programs. 

Saari's invention was limited to the application of ZA-worms (with straight-line profiles in the axial section) 
and the transformation of rotation between crossed axes only. The research of Litvin, Egelja, and 
De Donno was extended to the application of other types of worm thread surfaces and the transformation of 
rotation between intersected axes (in addition to the case of the crossed axes of rotation). 

Figures 2.7. 1 and 2.7.2 show respectively ( 1 ) a face worm-gear drive with a cylindrical worm and intersected 
axes of rotation and (2) a face worm-gear drive with a conical worm and crossed axes of rotation. 

We have to emphasize that pointing is much easier to avoid in face worm-gear drives in comparison with 
conventional face-gear drives because of the application of screw thread surfaces with different pressure angles 
for the driving and coast tooth sides and relatively small lead angle values. The width of the topland of a face 
worm-gear varies in a permissible range and does not equal zero. 

A detailed investigation to detect singularities based on the ideas presented in sections 1.4, 1.6, and 1.7 
enabled us to discover that the singular points on a face worm-gear tooth surface form an envelope E, to the 
contact lines that is simultaneously the edge of regression (fig. 2.7.3). Eliminating E, from the surface of the 
face gear guarantees the avoidance of undercutting. 

One of the important problems in designing face worm-gear drives is the localization of bearing contact and 
the predesign of a parabolic function of transmission errors. Based on the initial results of completed 
investigations, the authors of these research projects consider a promising solution to be the combination of 
profile and longitudinal deviations of the worm thread surface. These deviations are with respect to the 



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Figure 2.7.2.— Face worm-gear drive witli conical worm. 



2—. 




Figure 2.7.3. — Concave side of face worm-gear 
surface. (1) and (2) Surface brandies. (3) Contact 
lines. (4) Envelope to contact lines and edge of 
regression. 



theoretical worm thread surface that corresponds to the instantaneous line contact of worm and face worm-gear 
tooth surfaces. 

One of the great advantages of the face worm-gear drive is the increased strength of the worm in comparison 
with the spur pinion of a conventional face-gear drive. 



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2.8 Development of Cycloidal Gearing 

Introduction 

Cycloidal gears were initially used in watches. The profiles of these gears were represented as conventional 
and extended epicycloids and hypocycloids, the curves of which are traced out by the point of a generating circle 
of radius p that rolls over another circle of radius r(Litvin, 1994; Litvin and Feng, 1996). Circles of radii p 
and r may be in external and internal tangency. 

Cycloidal gearing has found new applications in Wankel engines, Root^s blowers, and screw compressors. 
In this section, these types of cycloidal gearing receive attention. 

Pin Gearing: Internal Tangency of Centrodes 

Figure 2.8. 1 shows that the gear centrodes are circles of radii r, and rj in internal tangency with point / being 
the instantaneous center of rotation. The pin of radius p is rigidly connected to centrode 1 . Curve I^ (an extended 
hypocycloid) is traced out in 52 by point C of the pin while centrode 1 is rolling over centrode 2. Coordinate 
systems 5, and ^2 (not shown in the figure) are rigidly connected to centrodes I and 2. Theoretically, point C 
and Z^can be considered conjugate profiles of links 1 and 2 of the gearing. In reality, the pin and curve X^, which 
is equidistant to X^, are used as conjugate profiles. Curves E, and I* are the two envelopes to the family of pins 
that are generated in coordinate system Sj, as seen in the example of figure 2.8.2. Another example of pin 
gearing (fig. 2.8.3) is based on the internal tangency of the envelope I* with the pin. 

Figures 2.8.4 and 2.8.5 show the schematic of the Wankel engine, which is based on the internal tangency 
of the envelope X/ with the pins. The chamber shape is determined by the envelope Z,* (see also fig. 2.8.1). 
The pins of the rotor may be connected as shown in figure 2.8.5. The connecting curve must be inside the space 
that is swept out in the chamber by the pins of the rotor, as shown in figure 2.8.6. The application of this pin 
gearing in Wankel's engine requires the avoidance of singularities in the engine chamber (Litvin and Feng, 
1997). 

Pin Gearing: External Tangency of Centrodes 

A particular case of pin gearing is the Root*s blower (Litvin, 1968; Litvin and Feng, 1996). The gear ratio 
is 1, and the two centrodes of the same radius r are in external tangency (fig. 2.8.7). The rotors are designed 
with two or three lobes. The "tooth" of the rotor is the combination of a pin that represents the addendum and 




Figure 2.8.1,— Generation of pin gearing by internal 
tangency of centrodes. 



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Link 2 



Centrode 2 




Link 1 



Centrode 1 



Figure 2.8.2.— Planar cycloidal gearing with external 
tangency of envelope 2^ and pin teeth. 




- — Centrode 1 



^ Link 1 

^ Centrode 2 
Figure 2.8.3.— Planar cycloidal gearing with internal 
tangency of envelope 2^ and pin teeth. 



Centrode 2 - 




2 (chamber) 



1^' ^ Centrode 1 

Figure 2.8.4.— Wankel's chamber and rotor. 



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- Chamber 



Rotor 




Connecting 
curve 



Figure 2.8.5.-"Wankel's engine. 




Figure 2.8.6.— Detemriination of space swept out in 
chamber by pins. 



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(b) 



Figure 2.8.7.— Root's blower, (a) With two lobes, (b) With three lobes. 



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2 (driven rotor) 



1 (driving 
rotor) 




Figure 2.8.8.— Cross sections of screw rotors. 



Figure 2.8.9.— Cross-sectional profiles of 
rotors 1 and 2. (a) Driven rotor, (b) Driving 
rotor. 



a concave curve that is conjugated to the pin of the mating rotor. The design requires that in a certain range the 
parameter ratio air be observed to avoid singularities of the dedendum profile (Litvin and Feng, 1996). 

Cycloidal Gearing of Screw Rotors of Compressors 

Screw compressors, in comparison with conventional compressors that are based on the application of a 
crank-slider linkage, enable the compressor to operate with an increased rotor angular velocity and to obtain 
a higher compression. However, the application of screw compressors requires that the rotor surfaces be 
conjugated (Litvin and Feng, 1997). 

The cross sections of screw rotors are shown in figure 2.8.8. The profiles of the rotors (fig. 2.8.9) are 
synthesized so that the rotor surfaces are in tangency at two contact lines. The synthesis is based on the following 
considerations: 

(1) The profile of driving rotor 1 (fig. 2.8.9(b)) is combined by curves a/') and by symmetrically located 
curves a/^) and ai'^\ Points .4, and A^ indicate the points of tangency of <T/>^ with cr/2)and a/^\ respectively. 

(2) Profile ai(» is designed as an elliptical curve, and a^^^KTxg. 2.8.9(a)) is the profile of the driven rotor that 
isconjugated to a/»). The meshing of the rotorsurfaces with profiles a/'^anda2t'>provides the first instantaneous 
line of tangency of the rotor surfaces. 

(3) Curves a^) and a^'^'^ of rotor 1 are epicycloids that are generated by points 5, and B-^ of the driven rotor 
(fig. 2.8.9). The second contact line of the rotor surfaces is the edge of rotor 2 (a helix of the screw surface of 
rotor 2). To reduce the wearing, the profile of rotor 2 may be rounded at the edge (Litvin and Feng, 1997). 



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PAGE INTENTIONALLY LEFT BLANK 



Chapter 3 

Biographies and History of Developments 

3.1 Introduction 

Visitors coming to the Gear Research Laboratory of the University of Illinois in Chicago are invited to see 
the Gallery of Fame, a photographic collection of gear company founders, inventors, and researchers who 
devoted their careers to the study and development of gears. This collection is unique because retrieving items 
and information for the gallery was a difficult task. Time had destroyed documents and memories of the many 
contributors who were deceased, requiring much detective work of the author who desired that these creative 
men be credited for their valuable contributions. In his effort to personalize these biographies, he sometimes 
had to find and then contact relatives, to search for information in libraries, and to communicate with gear 
company management to trace a predecessor's activities. The reader will have the opportunity to see, perhaps 
for the first time, the photographs of Samuel Cone, one of the inventors of the double-enveloping worm-gear 
drive; Ernst Wildhaber, one of the most successful inventors of gears; Dr. Chaim Gochman, the Russian scientist 
and founder of the gear theory; Dr. Hillel Poritsky, the American mathematician; Dr. Alexander Mohrenstein- 
Ertel, the founder of the hydrodynamic theory of lubrication; and many others who significantly contributed 
to this field. 

The author hopes that the biographies of these men will give the reader an opportunity to learn more about 
their lives through their thoughts, successes, and tribulations. 



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3.2 Johann Georg Bodmer — Inventor, Designer, and Machine Builder 
(1786-1864) 



The memorial plaque on the house at 12 Muhlegasse Street in Zurich, 
Switzerland bears the name of a former resident and distinguished engineer 
and inventor, Johann Georg Bodmer. We thank the Verein fiir 
Wirtschaftshistorische Studien of Switzeriand and R. Gardner (1941) for 
providing valuable information about the pioneering work of this remarkable 
man. Fortunately, Bodmer' s descendants, devoted to their famous forefa- 
ther, preserved his drawings and diaries for history. 

Johann Georg Bodmer came from a family of skilled handymen who 
traced their history in Switzeriand for more than 400 years. Members of this 
family were experienced stonecutters and textile workers. The young Bodmer 
graduated from the School of Art of Zurich when he was only 1 6, but he was 
highly motivated, talented, and creative and educated himself through 
experience. His inventions dealt with the textile industry, steam machines, 
locomotives, drives for ships, cannons, and various metal-cutting machines, 
including gear-cutting machines. Some of the machines that were designed 

for the production of tools used to manufacture screws and nuts were developed by successful cooperation with 

his son-in-law Jacob Reishauer, one of the founders of the Reishauer Company. 

Of special interest to gear engineers is that Johann Bodmer designed and manufactured spur gears with the 

ratio of 145 to 1 5, bevel gears, and helical gears. His achievements in the field of gears and in other areas of the 

machine industry were related to his remarkable ability to design tools. 

Undoubtedly, Johann Bodmer was in advance of his time and full recognition of his genius came later and 

not in his lifetime. 




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3.3 Friedrich Wilhelm Lorenz— Doctor of Engineering, h.c. and 
Medicine, h.c, Inventor, and Founder of the Lorenz Company 
(1842-1924) 




The life of Friedrich Wilhelm Lorenz is best described by the title of his 
biography Ein LebenfUrdie Technik (A Life for the Technique^ written by 
Dr. Susanne Pach-Franke (1990). Wilhelm Lorenz, an inventor and success- 
ful entrepreneur, was a self-made man who was bom in 1 842 in Geseke, a 
small German city. His father's position as court manager provided the 
family with very little income so that after his sudden death, Wilhelm could 
not continue his high school education and had to work when he was only 1 3 
years old. This biography covers the four stages of his career: ( 1 ) the first 
years of training, (2) his work for the military branch of industry, (3) the 
production of Daimler's motors, and (4) the founding of a company to 
manufacture gear production equipment. 

Wilhelm' s professional training began with his apprenticeship to a black- 
smith. His outstanding creativity was immediately revealed when he auto- 
mated the production of nails. Later, in 1 860, he obtained a position at Funcke 
and Hueck in Hannover where he was recognized as a gifted designer by the 
company's owner, Wilhelm Funcke, whose great affection for the young Lorenz paved the way for him to 
continue his professional training in Beriin. 

For a talented man, education is like a diamond cutter who awakens the brightness and shine of the jewel. 
Wilhelm Lorenz worked hard to educate himself and, not surprisingly, in 1870 was hired as an engineer by 
Georg Egerstorff of the Percussion Cap Company. There he learned the technological principles of the machines 
and tools used in the military branch of industry and automated high-precision manufacturing. In 1 875, he took 
a position at the Cartridge Case Company of Henri Ehrmann and Cie in Karlsruhe where after only 2 years he 
became the company's technical manager and then its owner. 

Although his career in industry was very successful, his desire to create led him to consider manufacturing 
car engines invented by Daimler. Lorenz' description of this period of his life was short and impressive: 
"Daimler invented motors for cars, but I, Lorenz, gave life to the cars." 

The last period of Lorenz' life will be particularly interesting to gear specialists. Initially, he was interested 
in and concentrated on the production of double-enveloping worm-gear drives and in 1891 invented methods 
to generate the worm and the gear of the drive. Having received two patents for this work, it was not a surprise 
that the worm-gear drive was his favorite invention and that, inspired by Julius Grundstein, Lorenz' company 
in Ettlingen specialized in the production of gear equipment. At the end of the 19th century, the company had 
become uniquely experienced in designing and producing precise metal-working machines, and in 1906 Julius 
Grundstein became its technical director. One of the first tests of the company's new direction was the design 
and production of a giant gear-cutting machine for gears having diameters up to 6 m, modules up to 100 mm, 
and tooth lengths up to 1 .5 m. 

In 1990, the Lorenz Company celebrated its 100th anniversary and is presently well known in the industry 
as a manufacturer of precise shaping machines for the production of external and internal involute gears, helical 
involute gears, chain-drive gears, and other special-profile gears (e.g., face gears) that can be generated by a 
shaper. The company also produces noncircular gears using CNC machines to execute the related motions of 
the shaper and the noncircular gear being generated (see ch. 12 in Litvin, 1994). 

Wilhelm Lorenz'achievements gained him the well-earned recognition of his contemporaries. In I9I0, 
Karlsruhe University awarded him an honorary degree of doctor of engineering, and Heidelberg University awarded 
him an honorary degree of doctor of medicine for his outstanding work in the manufacture of prostheses. 

Dr. Lorenz's life was darkened by several tragic events. His infant son and his young wife died from 
tuberculosis, and his other son was killed in an accident at age 12. He devoted his life to his daughter Ada and 
to his pioneering work. He is survived by Ada's two granddaughters. 

Although Dr. Lorenz did not want a gravestone and asked that his ashes be scattered by the wind, this 
remarkable man is remembered today in Geseke and Karlsruhe where streets bear his name and in Ettlingen 
where a school is named for him. However, the best memorial to his life in Ettlingen is his name on the company 
he founded. 



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3.4. Robert Hermann Pfauter — Inventor and Founder of the Pfauter 
Company (1854-1914) 



The end of the 1 9th and the beginning of the 20th century was the ''golden" 
era of pioneering developments in gear manufacturing. In 1897, Robert 
Hermann Pfauter invented the process and machine for generating spur and 
helical gears by hobs. Two other inventions were based on the application 
of shapers and rack-cutters (see sees. 3.5 and 3.6 on Erwin R. Fellows and 
Dr. Max Maag). These inventors were pioneers in creating gear-producing 
automotive machines based on the continuous relation of motions between 
the generating tool and the gear being generated. Such machines provide 
continuous indexing and generate the gear tooth surface as the envelope to 
the family of tool surfaces. 

Robert Hermann Pfauter was bom in 1 854 to a farm family in Goeltzschen, 
a village near Leipzig in Saxony. Fortunately for the future of the gear 
industry, he pursued technology rather than farming because of the prom- 
ising opportunities resulting from the industrial revolution. He began his 
professional career as a machinist and then continued his education at 




Hainichen Technical College near Chemnitz. He graduated with an engineering degree and went to work for 
the well-known machine tool companies Bimatzki and Zimmermann, Hartmann, and Reinecker. 

Pfauter* s employers soon recognized his talent and promoted him to higher positions, first as the principal 
design engineer at the Reinecker Company and then (at only age 39) as the technical executive director of the 
Chemnitz Knitting Machine Works (later known under the name of Bimatzki). 

The process of invention is a mystery and its crowning moment like a lightning bolt. For many years, Robert 
Hermann Pfauter noted the disadvantages of the existing manufacturing process for spur and helical gears: 
(1) the necessity of indexing the gear for the generation of each tooth space and (2) the generation of the tooth 
space only as a copy of the tool surface. His patent for generating spur and helical gears was free of these 
disadvantages and although it claimed to be limited to the generation of helical gears, it could actually be applied 
to the generation of spur and worm gears by hobs. In this respect, Pfauter was the precursor of a group of 
inventors who later proposed other gear generation processes based on principles similar to those proposed in 
his patent. 

Success in real life, unlike that in fairy tales, is a road of roses and thorns. It sounds trivial that there are no 
roses without thorns but this was true of Pfauter' s invention. He encountered many technical and financial 
obstacles while trying to prove his theories and was tempted to abandon his risky invention and go back to the 
safe work of a technical director. He decided otherwise, built the prototype of the machine in 1889, and finally 
in 1905 saw the Pfauter Company become a prosperous business. From its beginning to the spring of 1913, it 
manufactured 2(X)0 machines, many of which were imported to the United States. 

The engineering community recognized Pfauter as a brilliant man who designed, built, and produced for 
industry the machine he invented. He was remembered after his death in 1914 in an article in Chemnitzer 
Volksstimme (10/14/1914) not only as a distinguished inventor but also as an employer beloved by his 
coworkers. 

In the last 25 years, the Hermann Pfauter Company has become a multinational enterprise known as the 
Pfauter Group, serving the gear manufacturing community in many countries. At present it offers equipment 
for many gear manufacturing stages such as bobbing, shaping, hard skiving, shaving, honing, and milling. The 
company also produces cutting tools, noise-testing equipment, and a new generation of high-precision 
machines, including CNC gear grinding machines. The Pfauter Group's activity is proof that its founder 
successfully developed it. 



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3.5 Edwin R. Fellows — Inventor and Founder of the Fellows Gear 
Shaper Company (1865-1945) 



Edwin R. Fellows, another pioneer during the "golden" era of gear 
manufacturing, invented a revolutionary method to generate gears by shap- 
ing. His method is based on the simulation of two gears meshing during gear 
rotation about two parallel axes, which is in contrast to two other inventors 
who based their methods on hobs and rack-cutters (see sees. 3.4 and 3.6 on 
Robert Hermann Pfauter and Dr. Max Maag). 

The manufacture of spur gears by shaping originated at the Fellows Gear 
Shaper Company in Springfield, Vermont in 1896. Fellows' invention was 
a significant contribution because it enabled the automation of the internal 
and external gear manufacturing process. The automotive industry's broad 
application of the Fellows gear shaper process confirmed the importance of 
the invention. Continued development produced machines capable of gen- 
erating helical and face gears. 

In 1 897, the Fellows Company designed and built a machine for grinding 
involute profiles on gear shaper cutters. This process was later recognized for 

its application in grinding the various tooth profile cutters that were used to shape spur, helical, and special tooth 

profile gears. 

During the Second Worid War, the Fellows Company was awarded the distinguished Army-Navy 

Production Award for its outstanding contribution. Today, the Fellows Gear Shaper Company continues to 

prosper by designing and building innovative CNC machines with its latest Hydrostroke Gear Shaper, all of 

which attests to the importance of Edwin R. Fellows' pioneering innovations. 




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3.6 Max Maag — Doctor of Engineering, h.c, Inventor, and Founder of 
tlie Maag Company (1883-1960) 

The son of a village teacher in Flurlingen at Zurich, Max Maag attended 
the Zurich Polytechnicum but later decided to study machine design at the 
Oerlikon Company. Soon after, he became a designer at the Kaspar Wust 
plant. From 1908 to 1913, Dr. Max Maag's contributions helped to distin- 
guish him in the field of gearing. In 1908, he developed the geometry of 
nonstandard involute spur and helical gears, which enabled designers to 
avoid undercutting and to increase the tooth thickness by just modifying the 
installation of the tool (the rack-cutter) with respect to the gear being 
generated. His research led to the development of the Maag system of 
generating nonstandard involute gears. In this system, he applied the method 
and machine invented by Samuel Sunderland of Kieghley, York, in Great 
Britain and in 1 909 received the first patent for cutting nonstandard involute 
gears based on the application of Sunderland's machine. The advantage in 
generating gears by a rack-cutter was that it was a simpler and more precise 
tool than a hob and a shaper. 
The next 3 years were milestones in Maag's career. In 1910 he made a very courageous decision to open a 
small workshop to produce nonstandard involute gears with modified Sunderland's machines. The first gears 
cut according to the Maag method were delivered in 1911, and patents were granted in 14 countries. In 1913 
he founded the Maag Gear Wheel Company Ltd. and remained its managing director until his retirement in 1 927 
when the company was reorganized. Today, Maag machines have a reputation for quality and high-precision 
manufacturing and for the production of large-dimension gears. 

A significant achievement of the Maag Company was the development of the method to grind spur and 
helical involute gears. The grinding tool was a saucer-shaped wheel with a 15° profile angle, a prototype of 
which was built in 1913. This invention was important because it provided the first automatic compensation 
for the wear of the grinding wheel, making it the first and most famous automatic control system in the machine 
tool history. 

Several other contributions of the Maag Company made its name well known in the gear industry. In 1950, 
the method of grinding spur and helical gears by a plane was invented. The originality of the idea lay in the 
application of two grinding planes with a zero pressure angle, each plane simulating the surface of a rack tooth 
with a zero pressure angle. 

In recognition of his significant contributions to the design and manufacture of gears, the Swiss Federal 
Institute of Technology in Zurich awarded him an honorary doctorate in 1955. 

Dr. Maag devoted his life and creativity to investigating gears, as illustrated by his grave marker, which 
shows the meshing of a rack and an involute gear. 



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3.7 Earle Buckingham— Professor of Mechanical Engineering, Gear 
Researcher, and Consultant (1887-1978) 



Prof. Earle Buckingham is one of the founders of the theory of gearing and 
gear design and made significant contributions to this area. His monographs 
gained him international recognition in addition to the great respect he had 
already earned in English-speaking countries. 

Earle Buckingham was bom in 1887 in Bridgeport, Connecticut. He 
attended the United States Naval Academy in Annapolis, Maryland from 
1904 to 1906 and then began to work in industry. In 1919, he directed his 
research and science interests to the field of gears, initially as a gear 
consultant for the Niles-Bement-Pond Company (now Pratt & Whitney) and 
then from 1925 to 1954 as a professor of mechanical engineering at the 
Massachusetts Institute of Technology in Cambridge, Massachusetts. After 
retirement, he continued his research activity as a gear consultant. 

Prof. Earle Buckingham's books (e.g.. Analytical Mechanics of Gears, 
1 963) laid the foundation for the theory of gearing and became references for 
at least two generations of engineers and researchers. The engineering 
community recognized his contributions to the design and theory of gears by presenting him these prestigious 
awards: the American Society of Mechanical Engineers (ASME) Worcester Reed Warner Medal, 1944; the 
American Gear Manufacturers Association (AGMA) Edward P. Connel Award, 1950; the American Society 
of Test Engineers (ASTE) Gold Medal, 1957; the Gold Medal of the British Gear Manufacturers Association, 
1962; and the Golden Gear Award of Power Transmission Design magazine in commemoration of the AGMA 
50th Anniversary. Also in recognition of his contributions, Buckingham lectures are delivered at the USA 
Power Transmission and Gearing conferences. 




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3.8 Ernst Wildhaber— Doctor of Engineering, h.c, Inventor, and 
Consultant for The Gleason Works (1893-1979) 



Ernst Wildhaber is one of the most famous inventors in the field of gear 
manufacture and design. He received 279 patents, some of which have a 
broad application in the gear industry because of his work as an engineering 
consultant for The Gleason Works. Dr. Wildhaber' s most famous inven- 
tions are the hypoid gear drive, which is still used in cars, and the Revacycle 
method, a very productive way to generate straight bevel gears. 

Dr. Wildhaber graduated from the Technische Hochschule of Zurich 
University in Switzerland and then came to the United States in 1919. In 
1924, he went to work for The Gleason Works where he began the most 
successful period of his career as a creative engineer and inventor. 

Some of Wildhaber' s former colleagues from The Gleason Works have 
recounted how deeply impressed they were with his creativity, imagination, 
almost legendary intuition, and alacrity. This last characteristic was even the 
source of a colleague's complaint that Wildhaber was not as patient as 
university professors while giving explanations. Could it be that he expected 
his coworkers to comprehend at the same speed as his thoughts? They remembered that he even sped up the 
stairs, taking two steps at a time. 

Wildhaber' s inventions reveal signs of his originality. He proposed different pressure angles for the driving 
and coast tooth sides of a hypoid gear, which allowed him to provide constancy of the tooth topland. Wildhaber' s 
invention milestone was the Revacycle and the unusual shape of the tool based on the location of blades on a 
spatial curve. His theoretical developments (Wildhaber, 1946a, 1946b, 1946c, 1956) also display his original- 
ity; for example, he found the solution to avoiding singularities and undercutting in hypoid gear drives. 

Ernst Wildhaber' s talent was recognized not only in the United States but also by the worid engineering 
community and particularly by his alma mater, Zurich University, which awarded him an honorary doctorate 
in engineering in 1962. 




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3.9 Hillel Poritsky— Doctor of Mathematics, Professor of Applied 
Mechanics, and Consultant (1898-1990) 




A prominent mathematician. Dr. Hillel Poritsky made significant contri- 
butions to the fields of applied mathematics, the theory and geometry of 
gears, stress analysis, fluid dynamics, and electromagnetics. He authored or 
coauthored more than 100 scientific papers. One of his major interests was 
gear research, about which he and Darle Dudley wrote several papers 
dealing with the theory, geometry, and generation of worm-gear drives and 
helical gears. These pioneering works (e.g., Dudley and Poritsky, 1943) 
were ahead of their time and made a valuable contribution to gear research. 
Dr. Poritsky also conducted research on the hydrodynamic theory of gear 
lubrication. 

Hillel Poritsky is an example of an immigrant who came to the United 
States, pursued an education, devoted himself to his career, and contributed 
greatly to science and society. Bom in Pinsk, Belorussia, he lost both his 
parents before he was 5 and was raised by older sisters in conditions of 
poverty. He came to the United States when he was 15, not knowing a word 
of English and only able to order a meal in a cafe by pointing to the lowest priced items on the menu. However, 
he was a person of great intellectual ability and drive and became a distinguished scientist. 

After graduating from Morris High School in the Bronx in New York City, Dr. Poritsky went to Cornell 
University where his ability in mathematics and physics was soon recognized. Even before he received his 
bachelor's degree in 1920, he became an instructor. Also, during and after the First Worid War, he was on the 
staff of the United States Army Proving Ground in Aberdeen, Maryland where he used these abilities to solve 
problems. In 1927, he received his doctoral degree in mathematics from Cornell and in the same year 
went to Harvard as a National Research Council Fellow. Two years later in 1 929, he joined the General Electric 
Company in Schenectady, New York as a senior mathematician and consultant and worked there until he retired 
in 1963. 

Dr. Poritsky' s retirement mirrored his distinguished career in industry and academia. After he left General 
Electric, he worked as a consultant for the United States Navy and for several companies and was also adjunct 
professor at Rensselaer Polytechnic Institute in Troy, New York. In 1967, he was a visiting professor at the 
University of Illinois Department of Theoretical and Applied Mechanics. In this same year, his achievements 
in mathematics and applied mechanics were recognized by his contemporaries who awarded him the prestigious 
Timoshenko Medal, making him the first recipient. 

Dr. Poritsky was a member of several scientific societies: the American Society of Mechanical Engineers 
(ASME), the American Mathematical Society, and the American Institute of Electrical and Electronic 
Engineers. He was the chairman of the executive committee of the Applied Mechanics Division of the ASME. 



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3.10 Gustav Niemann— Professor and Doctor of Mechanical 
Engineering (1899-1982) 



Gustav Niemann was a prominent scientist and educator in the field of 
mechanism and machine design. He was bom on February 9, 1899 in 
RheineAVestfalen, Germany. From 1919 to 1923, he studied mechanical 
engineering at Darmstadt Technical University and received a doctor of 
engineering degree in 1928. He worked for industry from 1923 to 1934 and 
then held a professorship at Braunschweig University from 1934 to 1950. 
From 1951 to his retirement, Niemann spent the last period of his career 
at Munich Technical University where, as professor and head of the Institute 
of Machine Elements and the Gear Research Center, he conducted success- 
ful research and supervised 50 doctoral students. Many of these students 
later obtained important positions at universities and in industry where they 
disseminated and expanded Prof. Niemann' s ideas in Germany and through- 
out the world. 

He authored 150 scientific papers and a monograph on machine elements 

and held 50 patents. Among the valuable contributions of Prof . Niemann and 

his colleagues were the development of the concave-convex worm-gear drive (Niemann and Heyer, 1953), 

theoretical and experimental investigations of the power loss and load capacity of worm-gear drives, the friction 

of bearings, and the load capacity of gear materials. 

After Prof. Niemann retired, his research at the Institute of Machine Elements and the Gear Research Center 
was successfully continued first by Prof. Hans Winter and then by Prof. Bemd-Robert Hohn, who headed the 
Institute and then the Research Center. 

In recognition of Gustav Niemann's valuable contributions, the engineering and scientific communities 
presented him these awards: The Grashof Medal, the highest award of the VDI (Association of German 
Engineers); the E.P. Connell Medal, the prestigious award of the American Gear Manufacturers Association 
(Prof Niemann was the first non- American recipient); an honorary doctorate from Berlin Technical University; 
and an honorary membership in the Japan Society of Mechanical Engineers. 




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3.11 Meriwether L. Baxter, Jr.— Gear Theoretician (1914-1994) 




Meriwether L. Baxter was bom in 1914 in Nashville, Tennessee. He 
graduated from a Hartford, Connecticut public high school in 1931 and 
entered Yale University. After his graduation in 1935, he pursued his 
engineering and scientific work at The Gleason Works where he was the 
chief research engineer and the director of engineering until he retired in 
1975. He then worked as a consultant. 

In his early years at The Gleason Works, Meriwether Baxter spent time 
learning bevel and hypoid gear theory under Ernst Wildhaber's supervision, 
and Revacycle gears became his area of expertise. Then in the 1950*s, he 
recognized that the computer could be a powerful tool in the area of gear 
theory; it could relieve engineers of the long, tedious calculation procedures 
needed to make machine settings for generating bevel and hypoid gears. 
More important, he realized that with the computer and all its power, it was 
practical to apply vector and matrix notations in the development of gear 
theory and in doing so, he pioneered a new era in gear calculation and theory 

(Baxter, 1961, 1973). With respect to computers, his most noted accomplishments were developing tooth 

contact analysis (TC A), an undercut program and an exact duplex helical machine setting calculation procedure. 

He also held seven patents relating to gears and gear manufacturing. 

Meriwether Baxter was most concerned with education and training and spent much of his time ensuring 

that the methods and theory he developed be preserved and passed on to the next generation of gear engineers. 

To this end, Theodore Krenzer became his right-hand aid. All told, the contributions of Ernst Wildhaber, 

Meriwether Baxter, and Theodore Krenzer to the theory of gearing laid the foundation for important research 

that has continued successfully to the present. 

Meriwether Baxter was a member and fellow of the American Society of Mechanical Engineers and a 

member of the American Gear Manufacturers Association. 



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3,12 Hans Liebherr — Doctor of Engineering, h.c, Designer, and 
Founder of the Liebherr Group Companies (1915-1993) 



Hans Liebherr was a gifted designer and the founder of several successful 
industrial companies. He began his engineering career at his parents' 
construction company where he designed a crane that was mobile, portable, 
and easily assembled. This invention laid the foundation for the company he 
later established to produce various types of construction equipment. 
Eventually, Liebherr' s company became one of the largest of its kind in the 
world. 

Liebherr had a realistic perspective of technological advances and the 
talentto choose the right direction for his company. Forexample, he founded 
a division to produce aircraft equipment when he foresaw the future 
importance of aviation. In turn, this led him to the gear field and he founded 
a company that manufactured bobbing and shaping machines. Another 
example of his technological perspective was adding CNC grinding ma- 
chines to the product range in the late 1980's. 
In the early 1990's Liebherr took over Maschinenfabrik Lorenz, which 
manufactured gear shaping machines and cutters. The company's latest development is a new generation of 
CNC gear bobbing machines designed for hard metal and cermet tools for dry cutting. These machines have 
been very well received and put on the market. 

The great business success Hans Liebherr enjoyed did not change his personality. He remained a modest 
man, did not like publicity, and concentrated all his efforts on designing and producing gear equipment. His 
valuable contributions to German industry were recognized by Aachen University, which awarded him an 
honorary doctorate of engineering in 1964. 




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3.13 Alexander M. Mohrenstein-Ertel — Doctor of Engineering (1913-) 




Alexander Ertel-Mohrenstein made a tremendous contribution to the 
theory of hydrodynamic lubrication of loaded double-curvature surfaces. 
His theory has been applied to determine the lubrication conditions of gear 
tooth surfaces and bearings. 

Alexander Ertel (Oertel was the original German spelling) was bom to a 
German family who had deep roots in Russian society. His great grandfather 
Vasily Ertel, a doctor at Leipzig University, settled in Russia and became a 
citizen in 1 844. His descendants attained important positions in Russian 
society and became hereditary gentlemen (dvorjane). 

After graduation from high school, Alexander was admitted to the re- 
nowned Lomonosov University in Moscow in 1936 where he studied 
theoretical physics. His research in the hydrodynamic theory of lubrication 
was very successful, and his two fundamental papers, upon the recommen- 
dation of academician L.S. Leibenzon, were published in 1939 and 1945 in 
the Proceedings of the Academy of Sciences of USSR, Department of 
Technical Sciences. It is remarkable that his 1939 paper was published in the Proceedings while he was still a 
student. He graduated in 1941 and with the endorsement of the famous physicist Peter L. Kapiza, became a 
researcher at the Central Scientific Research Institute of Machine Industry (ZNIITMASH) in Moscow. While 
he worked at ZNIITMASH, Ertel prepared to defend his Russian doctoral thesis, which summarized the results 
of his research in 1945 and was presented in five seminars (1944-45) at the Institute of Machine Science of the 
USSR Academy of Sciences. Ertel' s thesis was highly acclaimed by the Russian reviewers. 

At the end of 1945, Dr. ErteFs life took a dramatic turn. While under assignment to the Soviet ministry, he 
visited Beriin-Pankow in East Germany (controlled by the Soviet Army) and decided to cross the border to the 
English zone and settle in West Germany. Had it not been for the prominent English scientist Dr. Alastair 
Cameron, who confirmed Ertel' s identity, he might not have been able to leave safely. Fortunately he did and 
began the second period of his research. 

In West Germany, Dr. ErteFs scientific work was initially performed under the name of Alexander von 
Mohrenstein and later under Mohrenstein-Ertel. From 1946 to 1948, working under the supervision of Prof. 
Gustav Niemann at the Technical University of Braunschweig, he defended the German version of his doctoral 
thesis in 1947. He also conducted research at the Max Plank Institute of Gettingen and at the University of 
Freiburg. 

A tragic car accident in 1959 interrupted Dr. Mohrenstein-Ertel's work; however, because of his wife's 
devotion and care, he has been able to continue in his private life. 



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3.14 Darle W. Dudley— Gear Consultant, Researcher, and Author 
(1917-) 




In some cases a person's contributions to a given field are not recognized 
by his contemporaries but by others long after his death. Fortunately, Darle 
W. Dudley did not have to wait for such recognition. In the field of gearing, 
he has gained worldwide recognition by writing books, acting as chairman 
of technical committees, writing standards for gear design, and promoting 
gear research. Some of his books are Practical Gear Design (1954), Gear 
Handbook (1962), The Evolution of the Gear Art (1969), Handbook of 
Practical Gear Design, (1984), and Dudley's Gear Handbook (edited by 
D.P. Townsend, 1991). 

Darle Dudley has written much about designing and manufacturing gear 
drives that are adequate for many applications. Having conducted numerous 
investigations of gear drives for large and small companies, he has become 
a "pathologist" in analyzing drives that prematurely failed. 

The Dudley family can trace its origin back more than 10 centuries to the 
small English town of Dudley where the ruins of a medieval castle give 
evidence of their past activities. From the standpoint of the Industrial Revolution, the castle is noteworthy 
because the first piston engine was put into service in a mine beneath it. 

Presently, thousands of Dudleys live in the United States and England. Darle Dudley descended from the 
Francis Dudley branch of the family who, in the first half of the 17th century during the English civil war, 
emigrated to Massachusetts seeking peace and religious freedom. 

Darle W. Dudley was bom on April 8, 1917 in Salem, Oregon. He graduated from Oregon State University 
in 1940 with a bachelor of science degree in mechanical engineering. He worked as an engineer and then as the 
manager of gear research and development at the General Electric Company (1940-64), as the manager 
of mechanical transmissions at Mechanical Technology, Inc. ( 1 964-67), and as the chief of gear technology at 
Solar Turbines Inc. (1967-78). In 1978, he took an early retirement and founded the Dudley Engineering Co., 
Inc. 

In recognition of his significant contributions to the development of gear art, Darle Dudley has received 
several awards in the United States and Europe. In 1958, the American Gear Manufacturers Association 
presented him the Edward P, Connell award. The most prestigious is the Ernst Blickle Prize awarded to him in 
Germany in 1995. 



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3.15 Oliver E. Saari— Inventor at the Illinois Tool Works (ITW) Spiroid 
Division (1918-) 




Oliver E. Saari is the creative and prolific inventor of Spiracon Roller 
Screws, Planoid Gears, Endicon, and new gear drives called Spiroid, 
Helicon, and Concurve (trademarks of ITW). Spiroid and Helicon gear 
drives transform rotation between crossed axes. The driving link of the 
Spiroid gear drive (Bohle and Saari, 1955; Saari, 1954) is a conical worm 
with a constant lead whereas the driving link of the Helicon gear drive 
(Saari, 1960) is a cylindrical worm. However, both these types differ 
substantially from conventional worm-gear drives: (1) the gear is a face 
type, and (2) the worm has different profile angles for the driving and coast 
sides. They offer specific advantages: one is that the worm and the gear are 
in line contact at every instant and the orientation of the line contact with 
respect to the relative velocity is favorable for the conditions of lubrication. 
This is especially important for drives with crossed axes where the sliding 
of surfaces is inevitable in all meshing zones. Another advantage is the 
almost constant top land of the gear helical teeth. The Concurve gear is a 
modification of the profiles of conventional involute spur and helical gears, which results in a constant profile 
curvature for the stabilization of contact stresses (Saari, 1972). Saari *s inventions bear the features of an 
unorthodox way of thinking which resulted in original ideas that have already been applied in industry and will 
be widely used in the future. 

The life and activity of Oliver Saari exemplify that the United States benefited as a country built by emigrants 
who valued education and worked exceedingly hard to attain it. He was bom on March 22, 1918, in Helsinki, 
Finland. In 1927, his family emigrated to the United States where, over a period of 8 years, they moved from 
Brooklyn, New York, to Rochester, Minnesota, and finally to Minneapolis. After graduation from high school, 
he joined the Civilian Conservation Corps (CCC) during which time he published several science fiction stories. 
He then learned tool and diemaking (his father's trade) at a vocational school before entering the University of 
Minnesota to study mechanical engineering. While at the university, he worked nights as a shop inspector at 
Honeywell Inc. Upon graduation in 1943, he went to work at the Buick Motor Division of General Motors 
Corporation in Flint, Michigan where he became interested in gears. In 1945, he joined the Illinois Tool Works 
(ITW) in Chicago and at the same time attended the Illinois Institute of Technology, receiving a master's degree 
in mechanics. In 1974, he retired from ITW and became a gear consultant. 



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Siiffiuel I. Cone 



3.16 History of the Invention of Double-Enveloping Worm-Gear Drives 

The invention of the double-enveloping worm-gear drive is a breathtaking 
story with two dramatic personae, Friedrich Wilhelm Lorenz and Samuel I. 
Cone, each acting in distant parts of the world — one in Germany and the 
other in the United States. 

Initially, the author intended to fmd a photograph of Samuel Cone ( 1 865- 
1949), the American inventor of the drive that carries the name of Cone 
Drive Co., the main producer of the aforementioned drives in the United 
States. Unfortunately, the photograph disappeared from the company files 
but was finally located in the family archives with the help of Cone's 
granddaughter Mrs. Mary Bell Kluge (nee Taylor). The search ended 
happily, but it was not the end of the story because during our visit to the 
Lorenz Co. in Ettingen, Germany, we learned that Dr. Lorenz had invented 
methods to generate the worm and the gear of the double-enveloping worm- 
gear drive and that he had received two patents for this work in 1891. 

Dr. Lorenz' invention was unknown to Samuel Cone, a modest draftsman 
to whom the idea of a double-enveloping worm-gear drive came indepen- 
dently, as seen in the following statement from the archives of the Cone 
Drive Co. in 1924: 

About 15 years ago, Mr. Samuel I. Cone of Portsmouth, 
Virginia, manufactured at the Norfolk Navy Yard the 
first Cone Type Worm. It is believed that this was the first 
instance in which Double Throated or Double 
Enveloping Worm Gears, having area contact, were ever 
made. 

Even though such worm-gear drives were manufactured at the Lorenz Co., 
the value of Cone's contribution cannot be diminished when one considers 
his courage and strong will. Cone acted as a lone inventor whereas Lorenz 
used the skills of his many employees. Still, it took a long time and required 
much effort before the Cone drive technology became applicable in the 
United States. We must pay credit to the Lorenz invention, but the father of 
the American invention was Samuel L Cone. Despite these facts, the 

question of who should claim exclusive rights for the double-enveloping worm-gear remains unanswered. Our 

opinion is that we have to credit both Lorenz and Cone, 

Wilhelm Lorenz and Samuel Cone understood very well the advantages of the drives they had invented, 

particularly, the increased load capacity due to the higher contact ratio in comparison with that of conventional 

worm-gear drives. Although the geometry of the Lorenz and Cone drives differs, both types offer this advantage. 

Later, investigations showed that the double-enveloping worm-gear drive's higher efficiency results from the 

existence of more favorable lubrication conditions. 

The complex geometry of the double-enveloping worm-gear drive, the specific conditions of lubrication 

(found later), and the formation of the worm-gear tooth surface as a two-part surface inspired many researchers 

to develop the analytical aspects of the meshing of the worm and the worm-gear tooth surfaces. Among such 

researchers are N. L Kolchin, B.A. Gessen, and P.S. Zak in Russia, Sakai in Japan, and Litvin in the United 

States, whose investigation and results are presented in Litvin (1968,1994). 

The primary manufacturer of double-enveloping worm-gear drives in the United States is the Cone Drive 

Textron Co. Presently, the Lorenz Co. does not produce these gear drives. 




Friedrich Wilhelm Lorenz 



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3.17 History of the David Brown Company 




Percx Brown 



Sir David Brown 



The David Brown story concerns a prosperous British company that 
originated in the Victorian era and was successfully run by one family for 
135 years. The company made significant contributions to the design and 
manufacture of gear transmissions and special products. 

In 1860, the first David Brown (1843-1903) founded the company when 
he was only 1 7. Originally, he began his business as a patternmaker for cast 
gears in Huddersfield and later extended it to the manufacture of cast gears. 
About 1898, his sons inspired him to begin manufacturing machine cut 
gears, an idea that later led to his making significant contributions to the 
design and manufacture of gear drives. 

After David Brown's death in 1903, his three sons, Ernest, Frank, and 
Percy, continued to run the company. Ernest took over the pattern business 
and Percy and Frank, the gear design and manufacture. 

When Percy Brown died in 1 93 1 and Frank in 1 94 1 , the company was then 
operated by David Brown (later Sir David), Frank^s son and the only 
grandson of the first David Brown. Significant engineering contributions 
made by the David Brown Company are connected with the names of these 
outstanding engineers who worked for the company: F.J. Bostock, Arthur 
Sykes, H.E. Merrit ( 1 97 1 ) (later doctor), and W. A.Tuplin (later doctor and 
professor). Among such contributions are the following: 

1. In 1915, the invention of an involute worm-gear drive based on the 
application of a screw involute surface as the surface of the worm 
thread (proposed by F. J. Bostock and Percy Brown) 

2. The ability to manufacture large gears (up to 40 ft in diam) following 
the design and installation in 1967 of a purposely built machine tool 
(later extended to 46 ft in diam with the acquisition of a Maag 1200) 

3. In 1933, the design and manufacture of the Radicon worm-gear drives 
(the trademark "Radicon" indicates heat dissipation by radiation, 
conduction, and convection due to the special design proposed by 
Arthur Sykes.) 

4. The successful design of vehicle transmissions (including the Merrit- 
Brown Tank Gearbox) 

5. The manufacture of high-precision gears used in turbine reducers and equipment designed for the 
production and inspection of gears 

In recognition of the David Brown Company's tremendous contribution to British industry during and after 
the Second Worid War, David Brown was knighted in 1 968, and Arthur Sykes received the Order of the British 
Empire (O.B.E.) and a special gold medal presented by the AGMA for his services to the industry. In addition, 
Arthur Sykes, Dr. Merrit, and Prof Tuplin gained wide recognition as engineering educators and textbook 
authors. Prof Tuplin became the head of the postgraduate Department of Applied Mechanics at Sheffield 
University. 

Sir David Brown could not perceive a logical successor to his family-operated company and sold it in 1 990, 
It continues its successful activity as the David Brown Group PLC under the management of Cris Cook and 
Cris Brown. 




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3.18 History of The Gleason Works 




William Gleason 



William Gleason (1836-1922), founder of The Gleason Works, came to 
the United States to Rochester, New York from Ireland in 1851. He was a 
gifted inventor and a skilled mechanic. In 1 874, his invention of the straight 
bevel gear planer for the production of bevel gears with straight teeth 
substantially advanced the progress of gear making. Manufacturing early 
straight bevel gears was a difficult process because the gears were cast in 
metal and then were finished by manually filing each tooth. 

Other members of the Gleason family made substantial contributions to 
the prosperity of The Gleason Works. Noteworthy are the inventions of 
James E. Gleason (see below) and the active role Miss Kate Gleason, the 
secretary and treasurer, played in the sale of the company's products. N. 
Bartels (1997) has written an impressive article about her life and activity 
in the journal Gear Technology. 

The early part of the 20th century was the beginning of the automotive 
industry, which required a broader application of bevel gears to transform 
rotation and power between intersected axes. In the 1920*s, automotive 
industry designers also needed (1) a gear drive to transform motions and 
power between crossed axes and (2) a lower location for the driving shaft. 
The Gleason Works engineers met these needs with pioneering develop- 
ments directed at designing new types of gear drives and the equipment and 
tools to generate the gears for these drives. The following patents granted to 
The Gleason Works proves how innovative their response was to the 
automotive industry's needs: 



1. In 1905, James E. Gleason invented a two-tool bevel generator that 
enabled manufacturers to reduce the production time of straight bevel 
gears by 2 times because two reciprocating blades could generate the 
opposite sides of the tooth space. 

2. In 1 9 1 6, the process for generating spiral bevel gears and the machine 
for producing them were developed. 

3. In 1927, The Gleason Works developed the method for generating 
hypoid gear drives. 

4. In 1938, James Gleason invented the Formate cut method by which spiral bevel gears and hypoid gears 
were generated. 




Kate Gleason 



At present. The Gleason Works holds over 700 patents. A historian studying its achievements would 
discover that the company's activity combines the design and production of cutting tools and equipment with 
intensive gear research. Its major developments in gear theory follow: 

1 . Spiral bevel gears and hypoid gears with the gear tooth surfaces being in point contact instead of in line 
contact: the bearing contact can be localized as a set of instantaneous contact ellipses, reducing the 
sensitivity of the gear drives to misalignment. 

2. Computerized methods (TCA, tooth contact analysis) that simulate the meshing and contact of hypoid 
and spiral gear drives: before manufacturing, these programs can be used to predict the shift in the bearing 
contact and the transmission errors caused by misalignment, allowing the design to be improved. 
Computer programs are commercially available for the gear industry. 

3. Coordinate measurement of gear tooth surfaces: the data obtained can be applied to correct machine tool 
settings. 



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4. Finite-element analysis is used to determine stresses and TCA programs to simulate the meshing and 
contact of loaded gear drives. 

5. Phoenix machines, six-axis, computer numerically controlled (CNC), for gear cutting and grinding: 
these machines can be used to modify the geometry of gears. 

Many of the solutions to gear manufacturing problems were contributed by Edward W. Bullok, Ernst 
Wildhaber(1926, 1946, 1956), Meriwether L.Baxter, Jr. (1961, 1973), Theodore J. Krenzer (1981), Lowe! E. 
Wilcox, Wells Coleman, and Dr. Hermann J. Stadtfeld, who also summarized the achievements of Gleason 
technology (Stadtfeld, 1993, 1995). 



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3.19 History of the Klingelnberg and Oerlikon Companies 




Em a Georg Biihrle 



Gust a V Adolj Klin^elnbe rg 



Klingelnberg Sohne and Oerlikon Geartec currently represent a group of 
companies very well known and respected for significant contributions that 
their engineers and managers have made to the technology and theory of 
gears, particularly, to the development of ( 1 ) the face-hobbed process of and 
the equipment for manufacturing spiral bevel gears and hypoid gears and (2) 
the method of grinding the worms of worm-gear drives (the worms are 
ground, or cut, by a disk whose surface is a cone). This method of 
manufacturing worms was applied by industry and resulted in a new type of 
worm-gear drives known as ZK-worms. 

The face-hobbed process of generating spiral bevel gears and hypoid gears 
is one of the crowning technological achievements in this area. Its great 
advantage is the efficiency of continuous indexing; its disadvantage is the 
difficulty of grinding (in comparison with the grinding of face-milled gears). 
The history of the developments by Klingelnberg and Oeriikon shows that 
the **crown" was not just a lucky finding but the result of enthusiastic efforts 
of the inventors and risky decisions of the company managers, as exempli- 
fied in the following discussions. 

In 1923, Klingelnberg Sohne built a machine that used the Palloid method 
for generating spiral bevel gears and hypoid gears with a conical hob. This 
method was invented in the 1920's by engineers Schicht and Preis and was 
later replaced by the face-milled and face-hobbed methods. 

The Oerlikon company began early, in 1 886, with the production of bevel 
gears by templates. Later developments are related to the names of inventors 
Brandenberger and Mamano and to Oerlikon-Biihrle Company engineers 
who designed the Spiromatic machines and the head-cutters for them. 

The Klingelnberg and the Oerlikon companies were successful because 
their founders were devoted and energetic. Among those we wish to mention 
are Dr. Gustav Adolf Klingelnberg and Emil Georg Biihrle. 

Under Dr. Khngelnberg's (1880-1947) leadership, Klingelnberg Sohne 
was transformed from a small trading company to a large manufacturer. In 
appreciation of his achievements and services to industry, Aachen Univer- 
sity awarded him an honorary doctorate. 
Emil Georg Biihrie (1 890-1956) was bom in Germany and in 1924 settled in Switzeriand where he became 
a naturalized citizen. His tremendous success there can be seen in his building the small Oeriikon Company to 
a group of more than 30 Oerlikon-Biihrle Companies located throughout the world. 




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3.20 History of the Reishauer Corporation 




Jacob Fried rich Reishauer 




Dorothea Reishauer 



The Reishauer Coiporation is presently well known in the gear industry for 
the development of methods and equipment for grinding high-precision 
involute spur and helical gears. These methods are based on the application 
of two types of grinding worms: ( 1 ) a cylindrical worm whose design 
parameters depend mainly on the normal pressure angle and the normal 
module of the gear to be ground and (2) a globoidal worm that is designed 
as the conjugate to the gear. The cylindrical worm and the gear being ground 
are in instant point contact whereas the globoidal worm and the gear are in 
line contact. Methods for dressing the grinding worms have been developed 
as well. 

These methods and machines are a logical extension of the skill and 
experience of a company that has been in existence for more than 200 years. 
The Reishauer Corporation was founded in 1788 by Hans Jakob Daniker, a 
blacksmith and toolmaker. The activity of the company was continued by 
Hans Daniker' s son-in-law Johann Gottfried Reishauer-Daniker (1791- 
1 848) whose name the company received in 1 824. This family operated the 
company over three generations: Jacob Friedrich Reishauer (1813-1862) 
represented the second generation and Georg Gottfried Reishauer (1840- 
1 885), the third. After 1 870, the ownership of the company was shared with 
other partners. 

The period when the company was operated by Jakob Friedrich Reishauer 
deserves special attention. Jakob, a gifted engineer, married Dorothea 
Bodmer, the daughter of Johann Georg Bodmer (1786-1864), a distin- 
guished inventor and engineer. Bodmer was well ahead of his time in 
developing tools to manufacture gears and may be considered the first 
pioneer in this area. The successful cooperation of Johann Bodmer and 
Jakob Reishauer resulted in their developing machines to produce the tools 
used in the manufacture of screws and nuts. They also designed and 
produced a lathe, a model of which still exists today. 

The history of the Reishauer Corporation illustrates that the company's 
engineers have always been successful in applying the skill and experience 
they accumulated throughout the years of its existence. 



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3.21 The Development of the Theory of Gearing in Russia 

Introduction 

An die Nachgeborenen, 
Wirklich, ich lehe infinstem Zeiten! 

To the Descendants. 
I live in dark times, indeed! 

In this verse, Bertolt Brecht (Brecht, 1962, p. 1 14), the convinced opponent of fascism, expressed with great 
emotion the pain he felt while living in a communist state in East Germany. Despite the Aesopian language, its 
meaning was clear for those who were unlucky enough to be born in the Soviet Union. Boris Pasternak, who 
described the same fear of communist terror in his novel Doctor Zhivago and was ostracized and persecuted 
for doing so, declared in the poem "The Nobel Prize" (1960, p. 246): 

/ am lost like a beast tracked down 

Somewhere men live in freedom and light 

But the furious chase closes in, 

And I cannot break out from my plight. 

For some who looked for a decent existence, safe harbor was found in their devotion to science. However, 
during the communist era in the former Soviet Union, all branches of science were controlled by party 
ideologues. The iron curtain separated Soviet scientists from the West, and the party falsified the history of 
developments in science and technique to further separate them. Admission to the graduate colleges and the 
recruitment of new researchers were controlled by the party through an insidious policy termed "the selection 
of personnel." 

Among those who tried not to obey the dirty policy of selection were Profs. Chrisanf F. Ketov, Nikolai I. 
Kolchin, Vladimir N. Kudriavtsev, and Lev V. Korostelev, to whom we are indebted for the development 
of the theory of gearing in Russia, We call the years 1946 to 1980, the golden years of the Russian theory of 
gearing, although the shells fell too close. A group of distinguished researchers formed the second generation 
of contributors to the theory of gearing: Profs. G. I. Sheveleva (1990, 1995), M. L. Erikhov (1966, 1996), 
Ya. S. Davidov (1950), V. L Goldfarb (1996), and Dr. S. A. Lagutin (1987, 1994), to name a few. 



100 NASA RP-1 406 



Chaim I. Gochman— Doctor of Applied Mathematics (1851-1916) 




The pre-Soviet period of the theory of gearing concerns the research of Dr. 
Chaim I, Gochman, a distinguished Russian scientist who deserves the title 
*the founder of the analytical theory of gearing." 

Theodore Olivier and Chaim Gochman, two great scientists, developed the 
foundation for the theory of gearing. Olivier introduced the idea of the 
generation of conjugate surfaces as the enveloping process and applied the 
concept of an auxiliary surface as the intermediate generating surface. In 
modem terms, the intermediate surface is called the tool surface. He also 
considered the gear tooth surfaces to be the envelopes to the auxiliary 
surface, and he discovered the way to provide the conditions for line contact 
and point contact of the generated surfaces of gears. However, Olivier 
insisted that the theory of gearing was exclusively the subject of projective 
geometry. 

Gochman, while confirming the indisputable contribution of Olivier, 
opposed his statement relating the theory of gearing to projective geometry. 
By developing analytical methods, he transferred the theory of gearing to analytical and differential geometry, 
releasing it from the constraints of projective geometry. His additional contribution was based on the possibility 
of determining the instantaneous line of contact of enveloping surfaces in any reference system, not only in the 
system of the driven gear as was determined previously. 

Gochman' s pioneering work was advanced for his time and was almost forgotten until Artobolevski, a 
Russian academician, "discovered" it again and inspired Prof. Kolchin (see below) to seek its application. 

Dr. Gochman's biography exemplifies one whose creativity is not recognized or acclaimed during his 
lifetime. He was bom in 1 85 1 in a small city in Grodnensk County, which at that time belonged to the western 
part of Russia. He graduated from Kherson gymnasium in 1871 when he was 20. His biographer Bogoliubov 
(1976) did not explain the reason for his late graduation from the gymnasium, but it was probably 
Dr. Gochman' s second education, the first being a Jewish theological one. He then attended Novorossiysk 
University in Odessa and graduated in 1 876, receiving the Golden Medal for his research project entitled 
"Analytical Method of the Solution to the Gearing Problems." He received a scholarship to continue his 
education at Novorossiysk University. In 1 88 1 , he passed the exams for a master' s degree and received a 2-year 
scholarship to study abroad. Gochman' s scientific career was proceeding well until, for some unknown reasons, 
he could not continue his activity at Novorossiysk University and had to work as an inspector for the Jewish 
Pedagogical Institute of Zhitomir. Here, he completed his master's thesis entitled "Theory of Gearing 
Generalized and Developed Analytically" (Gochman, 1 886). He lost his job in 1 887 when the Zhitomir Institute 
was closed, and he retumed to Novorossiysk University to work as an adjunct associate professor in the 
mechanics department. His second period of research activity began here. In 1890, Gochman was awarded a 
doctor's degree in applied mathematics for his work "Kinematics of Machines (Part 1)." Unfortunately, 
Dr. Gochman could not finish this work, planned as a capital research project. 



NASA RP-1406 



101 



Chrisanf F. Ketov — Doctor of Technical Sciences and Professor of 
Mechanisms and Machines (1887-1948) 




Chrisanf Ketov was bom to a family of peasants and had no hope of an 
education but that made possible by the heroic efforts of his widowed 
mother. In 1914, the young Ketov graduated from the highly ranked St. 
Peterburg Institute of Technology and received a postgraduate position with 
a track for teaching in the Department of Applied Mechanics. During the 
difficult postrevolutionary period, he left St. Peterburg in 1919 to take a 
professorship at Turkestan University. In 1 922, he returned to his alma mater 
as professor and head of the Department of Applied Mechanics. In 1930, 
after the colleges of two universities were joined, he became the head of the 
same department at the Leningrad Polytechnic Institute. He was then 
appointed dean of the College of Mechanics and Machines in 1944. He kept 
both positions until his death in 1948. One of his great contributions to 
education was in 1934 organizing, with Profs. Viakhirev and Kolchin, the 
Department of Automotive Machines. 
Prof. Ketov' s personal contributions to gear technology and theory in- 
clude a chapter in the Technical Encyclopedia ( 1927), his monograph on involute gearing ( 1 934), and a textbook 
on applied mechanics (in present terms, the theory of mechanisms and machines) written with Prof. Kolchin 
(1939). He is well known as a distinguished engineer and a recipient of several patents. He deserves a special 
tribute for supervising doctoral students' scientific work in various areas, especially in the theory of gearing. 
Prof Ketov' s role in the development of the theory of gearing should be recognized not only by his personal 
contributions but also by his influence on colleagues and students and by his founding The Leningrad 
Committee for Analysis of Gear Problems. The Leningrad Committee was soon recognized as the National 
Center of Gear Specialists and keeps the name of Prof Ketov in memory of his work. The author of this book 
worked for Prof Ketov for 1 1 years as an aide on the gear committee and as a professor in his department. 

Chrisanf F. Ketov was a self-made man whose knowledge of literature and history was extensive. He was 
considered in all circles a very intellectual man. He was a very impressive interlocutor who was able to enrich 
conversation with historical parallels and witty definitions. He had no illusions about the cultural environment 
and his lifetime and cited Leo Tolstoy who said, "There are moral people and immoral people but all 
governments are immoral." Chrisanf Ketov meant of course his own government. 

He faced difficult situations in his life. During the purge in the 1930's, he was arrested as a former "cadet" 
of the Constitution-Democrat Party, which was the most important liberal party in prerevolutionary Russia, but 
fortunately avoided the Gulag. Then as dean of the College of Mechanics and Machines, he was accused of 
resisting the policy of "selection of personnel," making his dismissal, if not his sudden death, inevitable. An 
ironic coincidence darkened the last day of his life when his request that a talented student be granted a position 
was declined because of the selection policy. 



102 



NASA RP-I406 



Nikolai I. Kolchin — Doctor of Technical Sciences and Professor of the 
Theory of Mechanisms and Machines (1894-1975) 




Profs. Ketov and Kolchin were colleagues for many years. They both 
graduated from St. Peterburg Institute of Technology and then worked for 
the Leningrad Polytechnic Institute. They were coauthors of a textbook on 
the theory of mechanisms and machines. It was not a surprise that very often 
their names appeared together and that they were considered a team. In one 
such instance at a party, toasts were proposed to both of them, and a slightly 
irritated Prof. Kolchin joked, "I do not mind if you join our names, but do 
not expect a wedding kiss." In reality, they had quite different personalities: 
Prof. Kolchin did not like administrative duties so the administrative job was 
usually accomplished by his aids out of respect for his many merits. 

Nikolai I. Kolchin was bom in 1894 in Sarapul to a lower middleclass 
family. After graduation from high school, he was admitted to the presti- 
gious St. Peterburg Institute of Technology . In 1 9 1 8 after graduation, he was 
assigned to a university position having a professorship track. Initially, his 
interests were in thermodynamics, and he published a few papers on this 
subject in highly ranked journals. However, later in 1939, he chose the theory of gears as his major scientific 
area. As professor and scientist, he concentrated his activities in the Leningrad Polytechnic Institute from 1935 
until his death in 1975 where he was the head of the department of mechanisms and machines. 

Prof. Kolchin made significant contributions to the theory of gearing; thus, it would be incorrect to consider 
him just a follower of Chaim Gochman. He developed the geometry of bevel gears and worm-gear drives with 
double-enveloping and cylindrical worms and published the monograph "Analytical Investigation of Planar 
and Spatial Gearing" (Kolchin, 1949), which became very well known. 

Prof Kolchin devoted his life to scientific research, not only during the week but on weekends and vacations. 
He supervised the programs of 50 doctoral candidates and doctors of technical sciences and was always 
available for his students and for those who needed to consult with him. 



NASA RP-1406 



103 



Mikhail L. Novilcov — Doctor of Technical Sciences and Professor and 
Department Head (1915-1957) 

Mikhail Novikov became very well known in his own country and in the 
international engineering community for the invention of a special type of 
helical gear. His idea was that he could overcome the barrier caused by the 
relations between the curvatures of the contacting surfaces when the gear 
tooth surfaces are in line contact. His approach is based on the following 
considerations: ( 1 ) the line contact of helical gear tooth surfaces is substi- 
tuted by point contact; (2) the unfavorable relations between the curvatures 
required for line contact of the tooth surfaces do not exist any more (because 
of point contact), but the contacting surface line of action must be parallel 
to the gear axes; and (3) the designer is free to choose a small difference 
between the curvatures to reduce the contacting stresses. 

Prof. Novikov proposed that the profiles of helical gear cross sections be 
chosen as circular arcs with a small difference between the radii of the circles 
(Novikov, 1956). Some researchers were reminded of Wildhaber, who 
proposed and patented a rack-cutter with a circular-arc profile (Wildhaber, 
1926), and they began to call Novikov gears Wildhaber-Novikov gears. The main difference between these two 
inventions is that Novikov gears are in point contact and have more favorable relations between the gear 
curvatures, Wildhaber helical gears are in line contact and the relations between the curvatures are constrained. 
Novikov' s invention became very popular in the former USSR and later in Russia, and he was awarded a 
prestigious national prize. His invention inspired many researchers, which resulted in valuable contributions 
being made to the theory of gearing. 

Mikhail Novikov graduated in 1940 from the Military Aircraft Engineering Academy that bears the name 
of the famous scientist Zhukovsky. He worked at the academy as a professor and department head and was 
involved in intensive research work, which unfortunately ceased with his sudden death at a young age, a 
catastrophe for his family, colleagues, and students. 




104 



NASA RP- 1406 




Vladimir N. Kudriavtsev — Doctor of Technical Science and Professor 
and Head of the Department of Machine Elements (1910-1996) 

Vladimir N. Kudriavtsev was recognized by his contemporaries as a 
distinguished scientist, teacher, and gear drive designer. He was remarkable 
in that his engineering intuition enabled him to predict the behavior of gear 
trains under load, a talent that impressed his colleagues. The following are 
among his distinguished contributions to this area: 

1 . Methods to synthesize planetary drives 

2. An effective approach to determine the efficiency of planetary trains 

3. Force distribution (a) along the instantaneous contact line, (b) between 
teeth in the case of a contact ratio larger than 1 , and (c) distribution of 
forces among satellites of a planetary train 

4. Development (as the leading author) of the USSR standard for comput- 
ing the strength of gears 

5. Excellent experimental tests to determine gear drive life service and 
failure 

The son of a physician, Vladimir Kudrjavtsev was bom in Ostrogozhsk, a small city in Voronezh county. 
After graduation from high school, he worked in the machine industry as a designer to acquire experience while 
he continued his education at the Leningrad Institute of Industry. He graduated in 1938 with distinction, 
receiving a bachelor of science degree in engineering. 

In 1942, he received a doctor's degree for his research entitled ^'Engineering Methods of Computation of 
Strength of Gear Trains." In 1952 he received the degree of doctor of technical science for his work 
"Investigation of Planetary Trains." 

In 1946, he began his teaching career as a professor of machine elements at the Leningrad Engineering 
Academy of the Air Forces and then in 1 958, as professor and department head of machine elements and director 
of the Laboratory of the Leningrad Institute of Mechanics. Books written by Prof. Vladimir N. Kudriavtsev and 
his colleagues became important handbooks for designers (Kudriavtsev, Kuzmin, and Filipenkov, 1993). Prof. 
Kudriavtsev published the well-known textbook Machine Elements and was an excellent teacher who 
supervised the scientific work of 67 doctoral candidates and 7 doctor of science programs. 

In recognition of his achievements. Prof. Kudriavtsev received the prestigious title "Honored Worker of 
Science and Engineering." 

In addition to his extensive scientific and engineering background, Prof Kudriavtsev was a passionate lover 
of music and literature as evidenced by his excellent taste and knowledge of both. 



NASA RP-1406 



105 



Lev V. Korostelev — Doctor of Technical Sciences and Professor 
(1923-1978) 

Lev V. Korostelev was born to a family who had been members of the 
intellectual class of Russian society for several generations. 

He graduated from the Moscow Machine and Tool Institute in 1950 and 
then concentrated his research work there. He received a Ph.D. in engineer- 
ing in 1954 and a doctor of technical sciences in 1964 for his thesis entitled 
''Geometric and Kinematic Criteria of Spatial Gearing Correlated with 
Loading Ability.'' He then successfully continued his research and made 
significant contributions, in recognition of which he was selected for the 
positions of department head and vice president of academic affairs of the 
university. 

Dr. Korostelev belonged to the second generation of scientists who 
developed the theory of gearing in Russia. He published approximately 100 
papers that dealt with the following subjects; the kinematic sensitivity of 
gears to alignment errors of the gear axes (Korostelev, 1970); the synthesis 
of spatial gearing by applying the concept of a generating surface performing 
screw motion; and the investigation of a generalized type worm-gear drive. He also received 30 patents and 
supervised the Ph.D. programs of 10 graduate students. 

To the deep regret of his relatives, friends, and colleagues, Dr. Korostelev' s very promising career was 
interrupted by his death in a car accident. 




106 



NASA RP-1406 



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NASA RP-1 406 107 



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and Machine Theory, Vol. 30, No. 8, pp. 1 171-1 178. 



NASARP-1406 109 



PAGE INTENTIONALLY LEFT BLANK 



Index 



Angular velocities, 19, 35, 38, 47, 50, 51 
Axes of meshing, 19, 21, 23, 28 
Axis of screw motion, 38, 40, 41, 44 
Axode, 38,41 

B 

Base circle, cylinder, 15, 54 

Bearing contact, 30, 31, 53, 59, 62, 63, 68, 71, 96 



Centrodes, 73 

CNC (computer numerically controlled) 

machines, 2, 81, 82, 83, 90, 97 
Common normal, 48, 52 
Common surface normal, 52 
Conjugation, 2 

Conjugation (of gear tooth surfaces), 1 
Conjugation of surfaces: 

rotor surfaces conjugated, 77 
Contact (of surfaces), instantaneous: 
line, 8,28,30,101,104 
point, 8,28,30,96, 101, 104 
Contact ellipse, 8, 9, 30, 96 
Contact line (see Line of contact, instantaneous), 

10,11,16,18,30,96,99,101 
Contact localization of, 28 
Contact normal (common normal to the contacting 

surfaces), 4, 23, 28, 47 
Contact path (see Path of contact), 30, 53, 56 
Contact point (current point of tangency of gear 

tooth surfaces), 4, 19, 23, 30, 47, 48, 96-97, 

104 
Contact ratio, 32,94, 105 



Crowning (of the gear tooth surfaces), 29, 59 
Curvature (normal, principal), 7 
Cycle of meshing, 32, 34 
Cylindrical worm, 49 

D 

Determinant, 23 

Discontinuous function of the transmission 

errors, 54 
Double-circular-arc helical gears, 32 
Double-enveloping worm-gear (drives), 94 

E 

Edge contact, 29, 32, 53, 56, 59, 62 

Edge of regression, 1 1 , ^2, 15,71 

Elastic deformation, 9, 30, 32 

Envelope, 3, 14,48 

Envelope (to family of surfaces, curves), 2, 3, 9, 

12, 48, 49, 82, 101 
Envelope to family of contact lines, 11, 16, 18 

on generated surface, 16, 28 

on generating surface, 18 
Enveloping, two-parameter, 28, 42, 49 
Equation of meshing, 2, 3, 5, 8, 14, 19, 20, 21, 

28,30,42,47,48,49,51 
Existence of envelope E2 to contact lines, 15 



Face gear (face-gear drive), 23, 28, 59, 60, 63, 

71,81,83 
Face worm-gear drives, 71 
Family of contact lines on generated surface, 1 1 
Family of generating surfaces, 1 4 
Family of surfaces, 2, 9, 28 



NASA RP-1406 



111 



Fillet, 6 

Flender worm-gear, 68 

Functions of transmission errors, 34 



Gearratio, 22,41,62,73 
Gearing: 

concurve, 1,93 

cycloidal, 73, 77 

involute, 1,53,56 

pin, 73 

planar, 1, 48 

spatial, 1, 12, 106 
Grinding disk, 28, 53, 62, 68, 98 
Grinding worm, 49, 54, 60, 99 

H 

Helical gears: 

involute, 12, 15, 23, 49, 56, 59, 80, 81, 82, 83, 
87, 93, 104 
Helicoid, 23, 28, 49 
Helicon gear drive, 93 

cylindrical, 71 

helicon, 93 
Helix, 15,45,46,48,56,59,77 
Hyperboloid of revolution, 41 
Hypoid gear (hypoid gear drive), 1, 29, 32, 41, 
53, 63, 86, 89, 96, 98 



Linear complex, 42, 46 

Local synthesis, 64 

Localization of contact, 28, 29, 53, 62, 68, 7 1 

M 

Matrix, 5, 11,14,23 

Meshing cycle (see Cycle of meshing), 8, 32 

Misaligned, 29, 30 

Misaligned helical gears, 56 

MisaHgnment, 28, 30, 32, 53, 54, 59, 62, 68, 71, 

96 
Mismatch (of tool surfaces), 29, 53, 63 
Mismatched generating surfaces, 63 
Modification, 32, 53, 56, 59, 63, 66, 68 

N 

Noncircular gears, 81 

Normal curvature, 5, 8 

Normal to surface, 4, 6 

Normal, unit normal to surface, 3, 9, 14, 18, 30, 

43, 47, 48, 52 
Null axis, 47, 48, 52 
Null plane, 46, 48, 52 

o 

Osculating plane, 45 
Oversized hob, 53, 68 



Instantaneous axis, 4 
Instantaneous axis: 

rotation, of, 60 

screw motion, of, 44, 66 
Instantaneous center of rotation, 4, 73 
Instantaneous line contact, 30, 72 
Interference, 8 
Involute gears 

helical (see Helical gears, involute), 7, 30, 
81,84 

spur, 82, 83, 84, 93, 99 
Involute gears, 84 



Jacobian, 29,31,56 

L 

Line contact (see Contact of surfaces, instanta- 
neous), 2, 8, 28, 29, 53, 56, 66, 68, 93, 104 
Line of action, 20, 30, 35, 37, 104 
Line of contact, instantaneous, 15, 30, 48, 101, 
105 
line contact at every instant, 48 



Parabolic linear, 34 

Path of contact, 29, 59 

Pitch surfaces, 60 

Pitch: 
cones, 41, 60 
cylinders, 41,49,50,60 

Planar gearing, 2, 4 

Pluecker^s coordinates, 42, 44 

Pluecker's linear complex, 42, 43, 44, 46, 48, 51, 
52 

Plunging, 53, 62 

Point contact, 8, 29, 30, 53, 66, 104 

Point of tangency, 52 

Pointing, 60,71 

Position vector, 13, 19, 22, 29, 42, 48, 51 

Predesigned function of transmission errors (see 
Transmission errors: predesigned parabolic 
function, of), 32, 53, 54, 56, 59, 64, 71 

Principal curvatures, 5, 8 



Rack, rack-cutter, 2, 6, 12, 54, 59, 68, 82, 83, 84, 

104 
Regular surface, 2, 5, 9, 1 1 



NASA RP-1 406 



112 



Relative velocity, 4, 47, 48, 5 1 
Revacycle method, 1, 86, 89 
Root's blower, 73 



Screw compressors, 73, 77 

Screw motion, 4, 28, 38, 40, 41, 42, 44, 46, 48 

Screw parameter, 15, 23, 28, 38, 40, 44, 51, 66 

Shaper, 2,60,81,82,83,84 

Simulation of meshing, 28, 29, 30, 31, 59, 66, 68, 

83 
Singularities, 15,71,73 
Singularity, singular point, 5,7, 11, 18,71,86 
Spiral bevel gears, 29, 53, 63, 64, 96, 98 
Spiroid gear drive, 23, 7 1 , 93 
Spur gear, 7, 15,53 

Sufficient conditions for existence of envelope, 9 
Surface normal, 5, 21 
Surface of action, 10 
Surface representation, 14, 16 
Surface torsions, 5 



Tangent, 10, 12, 16, 17 

Tangent plane, 3, 8, 30, 48 

Tangent, unit tangent, 4, 48 

TCA (tooth contact analysis), 29, 30, 63, 71, 89, 

96,97 
Theorems: 

implicit function system existence, 3, 11, 17, 

29 
Zalgaller's, 2,9,11 
Tip of the contact normal, 4 
Torque split, 59 
Torsions, 8 
Transfer from line contact of the surfaces to 

point, 30 
Transfer of meshing, 32 
Transition from surface line contact to point 

contact, 30 
Transition point on a contact line, 3 1 
Transmission errors: 32, 34, 53, 54, 59, 63, 64, 

66,71,96 



discontinuous linear function, of, 34 
nonlinear, 32 

predesigned parabolic function, of, 32, 34, 64 
Two-parameter enveloping, 28, 42, 49 

u 

Undercutting (nonundercutting), 5, 8, 60, 71, 84, 

86 
Unit normal, 4, 19,22,23,29 
Unit vector, 9, 14,45,48 



Vector function, 2, 51 
Vector: 

free, 35 

moment, 19, 35, 38, 44, 47, 48 

position, 35, 37, 44 

sliding, 4, 28, 35, 37 

unit, 4 
Velocity, 12,44,46,51 
Velocity: 

angular relative, 4, 35, 38, 44 

relative, 4, 93 

relative (sliding), 3, 4 

relative (sliding), of a point of one surface 
with 47 

tip of contact normal, 4 

w 

WankeFs engine, 73 

Worm-gear, 68, 94 

Worm-gear drives, 28, 30, 41, 66, 68, 71, 81, 87, 

88,93,94,95,98,103,106 
Worm-gear drives, double-enveloping, 79, 81, 94, 

103 
Worm: 60 

ZA (Archimedes'), 66,71 

ZF (Flender), 66 

ZI (Involute), 66 

ZK (KUngelnberg), 66,98 

ZN (Convolute), 66 



NASARP-1406 



«0.S. OOVEraWENT PRIMrrNG OFFICE: 1998-651-507 



113 



REPORT DOCUMENTATION PAGE 



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Public reporting burden for this collection of intormation is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 
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1. AGENCY USE ONLY (Leai^e Wan/t) 



REPORT DATE 

December 1997 



3. REPORT TYPE AND DATES COVERED 

Reference Publication 



4. TITLE AND SUBTITLE 



Development of Gear Technology and Theory of Gearing 



6. AUTHOR(S) 

Faydor L. Litvin 



PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

NASA Lewis Research Center 

Cleveland, Ohio 44135-3191 

and 

U.S. Army Research Laboratory 

Cleveland, Ohio 44135-3191 



5. FUNDING NUMBERS 



WU-581-30-13 
1L162211A47A 



8. PERFORMING ORGANIZATION 
REPORT NUMBER 



E- 10679 



9. SPONSORING/MONITORING AGENCY NAME{S) AND ADDRESS(ES) 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

and 

U.S. Army Research Laboratory 

Adelphi, Maryland 20783-1145 



10. SPONSORING/MONITORING 
AGENCY REPORT NUMBER 

NASARP-1406 
ARL-TR-1500 



11. SUPPLEMENTARY NOTES 

Faydor L. Litvin, University of Illinois at Chicago, Department of Mechanical Engineering, Chicago, Illinois 60607-7022. 
Responsible person, Robert F. Handschuh, Army Research Laboratory, Vehicle Technology Center, NASA Lewis 
Research Center, Structures and Acoustics Division, organization code 5950, (216) 433-3969. 



12a. DISTRIBUTION/AVAILABILITY STATEMENT 

Unclassified - Unlimited 
Subject Category: 37 



Distribution: Standard 



This publication is available from the NASA Center for AeroSpace Information, (301) 621-0390. 



12b. DISTRIBUTION CODE 



13. ABSTRACT (Maximum 200 words) 

This book presents recent developments in the theory of gearing and the modifications in gear geometry necessary to improve 
the conditions of meshing. Highlighted are low-noise gear drives that have a stable contact during meshing and a predesigned 
parabolic transmission error function that can handle misalignment during operation without sacrificing the low-noise aspects 
of operation. This book also provides a comprehensive history of the development of the theory of gearing through biographies 
of major contributors to this field. The author* s unique historical perspective was achieved by assiduous research into the lives 
of courageous, talented, and creative men who made significant contributions to the field of gearing. 



14. SUBJECT TERMS 

Gear geometry; Gears; Power transmission; Transmissions 



15. NUMBER OF PAGES 

124 



16. PRICE CODE 

A06 



17. SECURITY CLASSIFICATION 
OF REPORT 

Unclassified 



18. SECURITY CLASSIFICATION 
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Unclassified 



19. SECURITY CLASSIFICATION 
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Unclassified 



20. LIMITATION OF ABSTRACT 



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